# Small-Step Walks: Closed Forms for Generating Functions

In all formulas below, all functions (whether rational, algebraic, or ${}_{2}F_{1}$) have to be understood as formal Laurent series, and integration $\int$ is the operator defined by $\displaystyle \int A(t) \, {\rm d}t = \sum_{n \geq m} a_n \frac{t^{n+1}}{n+1}$ for $\displaystyle A(t) = \sum_{n \geq m} a_n t^n$, provided $a_{-1} = 0$ (which always holds in the list below). An alternative, more analytic, representation would be $\displaystyle \int A(t) \, {\rm d}t = - \int_t^\infty \operatorname{pp}(A)(u) \, {\rm d}u + \int_0^t (A - \operatorname{pp}(A))(u) \, {\rm d}u$ after defining $\displaystyle \operatorname{pp}(A)(t) = \sum_{n < 0} a_n t^n$.

ssvxvyoccurring ${}_{2}F_{1}\!(a,b;c;w)$, standardizedclosed form
01 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle 1/2\,{\frac {1-{{}_{2}F_{1}\!\left(-1/2,1/2;2;16\,{t}^{2}\right)}}{{t}^ {2}}} 01\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/4\,{\frac {1-2\,t-{{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}+ 2\,t{{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{t}^{2}\right)}}{{t}^{2}}}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle 1/4\,{\frac {1-2\,t-{{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}+ 2\,t{{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{t}^{2}\right)}}{{t}^{2}}} 11\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/2\,{\frac {{{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}-1}{t}}+ {{}_{2}F_{1}\!\left(1/2,3/2;2;16\,{t}^{2}\right)}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{ x}^{-1} \right) ^{2}{t}^{2}}\int \!{t \left( 2+\int \! \left( 2\,t+2\,{ \frac {t}{{x}^{2}}}-3\,{x}^{-1} \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{t}^{2}\right)}\,{\rm d}t \right) \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{ 2}{t}^{2} \right) ^{-3/2}}\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( y-{ y}^{-1} \right) ^{2}{t}^{2}}\int \!{t \left( 2+\int \! \left( 2\,t+2\,{ \frac {t}{{y}^{2}}}-3\,{y}^{-1} \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{t}^{2}\right)}\,{\rm d}t \right) \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y-{y}^{-1} \right) ^{ 2}{t}^{2} \right) ^{-3/2}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}y+x{y}^{2}+x+y \right) } \left( xy-{ \frac {x}{t}\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{2}{t}^{2}}\int \!{t \left( 2+\int \! \left( 2\,t+2\,{\frac { t}{{x}^{2}}}-3\,{x}^{-1} \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{t}^{2}\right)}\,{\rm d}t \right) \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{ 2}{t}^{2} \right) ^{-3/2}}\,{\rm d}t}-{\frac {y}{t}\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( y-{y}^{-1} \right) ^{2}{t}^{2}}\int \!{t \left( 2+\int \! \left( 2\,t+2\,{\frac {t}{{y}^{2}}}-3\,{y}^{-1} \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{t}^{2}\right)}\,{\rm d}t \right) \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y-{y}^{-1} \right) ^{2}{t}^{2} \right) ^{-3/2}}\,{\rm d}t} \right) } 02 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/4\,{\frac {{{}_{2}F_{1}\!\left(-1/2,-1/2;1;16\,{t}^{2}\right)}-1}{{t} ^{2}}}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{t}^{2}\right)} 10\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{t}^{2}\right)}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle \left( 1+1/4\,{t}^{-1} \right) {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}-1/4\,{t}^{-1} x0\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle {\frac {\sqrt {1-4\, \left( x+{x}^{-1} \right) ^{2}{t}^{2}}}{{t}^{2}} \int \!{\frac {t \left( 2+ \left( 1+{x}^{-2} \right) \left( {{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}-1 \right) \right) }{ \left( 1-4\, \left( x+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}}} \,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle {\frac {\sqrt {1-4\, \left( y+{y}^{-1} \right) ^{2}{t}^{2}}}{{t}^{2}} \int \!{\frac {t \left( 2+ \left( 1+{y}^{-2} \right) \left( {{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}-1 \right) \right) }{ \left( 1-4\, \left( y+{y}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}}} \,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}+1 \right) \left( {y}^{2}+1 \right) } \left( xy-{\frac { \left( {x}^{2}+1 \right) \sqrt {1-4\, \left( x+{x}^ {-1} \right) ^{2}{t}^{2}}}{t}\int \!{\frac {t \left( 2+ \left( 1+{x}^{- 2} \right) \left( {{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}-1 \right) \right) }{ \left( 1-4\, \left( x+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}}}\,{\rm d}t}-{\frac { \left( {y}^{2}+1 \right) \sqrt {1 -4\, \left( y+{y}^{-1} \right) ^{2}{t}^{2}}}{t}\int \!{\frac {t \left( 2+ \left( 1+{y}^{-2} \right) \left( {{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}-1 \right) \right) }{ \left( 1-4\, \left( y+{y}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}}} \,{\rm d}t}+1/4\,{\frac { {{}_{2}F_{1}\!\left(-1/2,-1/2;1;16\,{t}^{2}\right)}-1}{t}} \right) }$
03 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} 01\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+ 1 \right) ^{5/2}} \left( \left( 12\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -3\,t \left( 6\,t-1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+ 1 \right) ^{5/2}} \left( \left( -20\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} +20\,{t}^{2} {{}_{2}F_{1}\!\left(3/4,9/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {t}{ \left( 2\,t+1 \right) \left( 6\,t+1 \right) }}\right)}$$\displaystyle {\frac {1}{t}\int \!{\frac {1-2\,t}{ \left( \left( 2\,t+1 \right) \left( 6\,t+1 \right) \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/2,3/2;2;16\,{\frac {t}{ \left( 2\,t+1 \right) \left( 6\,t+1 \right) }}\right)} }\,{\rm d}t}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+ 1+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2+\int \!24\,{ \frac {t\sqrt {1-4\, \left( x+1+{x}^{-1} \right) ^{2}{t}^{2}}}{{x}^{2} \left( 12\,{t}^{2}+1 \right) ^{7/2}} \left( 20\, \left( {x}^{2}+x+1 \right) {t}^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 1+x \right) ^{2} \left( 12\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( \left( 1+ \left( y+{y}^{-1} \right) t \right) \left( 1-3\, \left( y+{y}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt { \left( 1+ \left( y+{y}^{-1} \right) t \right) \left( 1-3\, \left( y+{ y}^{-1} \right) t \right) }}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{y}^{2 }} \left( 60\, \left( {y}^{2}+1 \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( 3\,t{y}^{2}+3\,t+y \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}+x+1 \right) \left( {y}^{2}+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+1+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2+ \int \!24\,{\frac {t\sqrt {1-4\, \left( x+1+{x}^{-1} \right) ^{2}{t}^{2 }}}{{x}^{2} \left( 12\,{t}^{2}+1 \right) ^{7/2}} \left( 20\, \left( {x} ^{2}+x+1 \right) {t}^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 1+x \right) ^{2} \left( 12\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+1}{ t}\int \!\!\!\int \!{\frac {1}{ \left( \left( 1+ \left( y+{y}^{-1} \right) t \right) \left( 1-3\, \left( y+{y}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt { \left( 1+ \left( y+ {y}^{-1} \right) t \right) \left( 1-3\, \left( y+{y}^{-1} \right) t \right) }}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{y}^{2}} \left( 60\, \left( {y}^{2}+1 \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( 3\,t{y}^{2}+3\,t+y \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac {{t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) } 04 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{t} \left( -1- \left( 2-{t}^{-1} \right) \int \!{\frac {1}{ \left( 2\,t-1 \right) ^{2}} {{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} }\,{\rm d}t \right) }$
01$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 4\,t+1 \right) ^{4}} \left( {{}_{2}F_{1}\!\left(3/2,3/2;2;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} + {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} t \right) }\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 4\,t+1 \right) ^{4}} \left( {{}_{2}F_{1}\!\left(3/2,3/2;2;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} + {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} t \right) }\,{\rm d}t\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{t}\int \!{\frac {1}{ \left( 4\,t+1 \right) ^{3}} {{}_{2}F_{1}\!\left(3/2,3/2;2;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} }\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( x+ {x}^{-1} \right) t- \left( 3\,{x}^{2}+2\,x+3 \right) \left( 1+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2-6\,{\frac {1}{{x}^{2}} \int \!{\frac {\sqrt {1-2\, \left( x+{x}^{-1} \right) t- \left( 3\,{x}^ {2}+2\,x+3 \right) \left( 1+{x}^{-1} \right) ^{2}{t}^{2}}}{ \left( 4\, t+1 \right) ^{5}} \left( \left( 4\,t+1 \right) \left( \left( 3\,{x}^ {2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(3/2,5/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} -12\,{t}^{2} \left( {x}^{2}+x+1 \right) {{}_{2}F_{1}\!\left(5/2,5/2;4;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t} \right) }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( y+ {y}^{-1} \right) t- \left( 3\,{y}^{2}+2\,y+3 \right) \left( {y}^{-1}+1 \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2-6\,{\frac {1}{{y}^{2}} \int \!{\frac {\sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^ {2}+2\,y+3 \right) \left( {y}^{-1}+1 \right) ^{2}{t}^{2}}}{ \left( 4\, t+1 \right) ^{5}} \left( \left( 4\,t+1 \right) \left( \left( 3\,{y}^ {2}+4\,y+3 \right) t+y \right) {{}_{2}F_{1}\!\left(3/2,5/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} -12\,{t}^{2} \left( {y}^{2}+y+1 \right) {{}_{2}F_{1}\!\left(5/2,5/2;4;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t} \right) }\,{\rm d}t\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}y+x{y}^{2}+{x}^{2}+{y}^{2} +x+y+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!\!\!\int \!{ \frac {1}{ \left( 1-2\, \left( x+{x}^{-1} \right) t- \left( 3\,{x}^{2}+ 2\,x+3 \right) \left( 1+{x}^{-1} \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2-6\,{\frac {1}{{x}^{2}}\int \!{\frac {\sqrt {1-2\, \left( x+{x }^{-1} \right) t- \left( 3\,{x}^{2}+2\,x+3 \right) \left( 1+{x}^{-1} \right) ^{2}{t}^{2}}}{ \left( 4\,t+1 \right) ^{5}} \left( \left( 4\,t +1 \right) \left( \left( 3\,{x}^{2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(3/2,5/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} -12\,{t}^{2} \left( {x}^{2}+x+1 \right) {{}_{2}F_{1}\!\left(5/2,5/2;4;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t} \right) }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+y+ 1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}+2\,y+3 \right) \left( {y}^{-1}+1 \right) ^{2}{t}^{2} \right) ^{3/2}} \left( 2-6\,{\frac {1}{{y}^{2}} \int \!{\frac {\sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^ {2}+2\,y+3 \right) \left( {y}^{-1}+1 \right) ^{2}{t}^{2}}}{ \left( 4\, t+1 \right) ^{5}} \left( \left( 4\,t+1 \right) \left( \left( 3\,{y}^ {2}+4\,y+3 \right) t+y \right) {{}_{2}F_{1}\!\left(3/2,5/2;3;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} -12\,{t}^{2} \left( {y}^{2}+y+1 \right) {{}_{2}F_{1}\!\left(5/2,5/2;4;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t} \right) }\,{\rm d}t\,{\rm d}t}-1- \left( 2-{t}^{- 1} \right) \int \!{\frac {1}{ \left( 2\,t-1 \right) ^{2}} {{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{\frac { \left( t+1 \right) t}{ \left( 4\,t+1 \right) ^{2}}}\right)} }\,{\rm d}t \right) }$
05 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {\int \!2\,t{{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)} \,{\rm d}t}{{t}^{2}}} 01\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle {\frac {1}{ \left( t-1 \right) t}\int \!{\frac {6\,{t}^{3}+2\,{t}^{2}+t -1}{\sqrt {-3\,{t}^{2}-2\,t+1}{t}^{2}}\int \!{\frac {t \left( 16\,{t}^{ 3}+6\,{t}^{2}-6\,t+1+2\,{t}^{2} \left( 22\,{t}^{4}+6\,{t}^{3}-17\,{t}^{ 2}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}- \left( 44\,{t}^{4}-16\,{t}^{2}+6\,t-1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} \right) }{ \left( 6\, {t}^{3}+2\,{t}^{2}+t-1 \right) ^{2}\sqrt {-3\,{t}^{2}-2\,t+1}}} \,{\rm d}t}\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {-8\,{t}^{2}+1}} \left( 2- \int \!8\,{\frac {t{{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{ \sqrt {-8\,{t}^{2}+1}}}\,{\rm d}t \right) }\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle {\frac {1}{ \left( t-1 \right) t}\int \!{\frac {{t}^{2} \left( t+1 \right) }{ \left( -3\,{t}^{2}-2\,t+1 \right) ^{3/2}} \left( -7+\int \! {\frac {\sqrt {-3\,{t}^{2}-2\,t+1} \left( 1+ \left( -10\,{t}^{3}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;1;64\,{t}^{4}\right)}+6\,{t}^{3} \left( 14\,{t}^{2}-8\,t+3 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{t}^{4}\right)} \right) }{{t}^{3} \left( t+1 \right) }}\,{\rm d}t \right) }\,{\rm d}t}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}x}}\,{\rm d}t \right) }\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle {\frac {1}{t \left( t-y \right) }\int \!{\frac {8\,{t}^{3}{y}^{4}-2\,{t }^{3}+2\,{t}^{2}y+t{y}^{2}-{y}^{3}}{{t}^{2}}\int \!{\frac {t}{ \left( 8 \,{t}^{3}{y}^{4}-2\,{t}^{3}+2\,{t}^{2}y+t{y}^{2}-{y}^{3} \right) ^{2}} \left( {y}^{2} \left( 16\,{t}^{3}{y}^{3}+6\,{t}^{2}-6\,ty+{y}^{2} \right) +2\,{t}^{2} \left( {y}^{2}-5\,{t}^{2}-12\,{y}^{4}{t}^{2}+6\,{ \frac {{t}^{3}}{y}}-2\,{\frac {{t}^{4}}{{y}^{2}}}+24\,{y}^{2}{t}^{4} \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}- \left( 48\, {t}^{4}{y}^{4}-16\,{t}^{3}{y}^{5}-4\,{t}^{4}+16\,{t}^{3}y-16\,{t}^{2}{y }^{2}+6\,t{y}^{3}-{y}^{4} \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} \right) {\frac {1}{ \sqrt {1-2\,{\frac {t}{y}}- \left( 4\,{y}^{2}-{y}^{-2} \right) {t}^{2}} }}}\,{\rm d}t{\frac {1}{\sqrt {1-2\,{\frac {t}{y}}- \left( 4\,{y}^{2}-{ y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{y}^{2}+x \right) } \left( xy-{ \frac {x}{t}\int \!{\frac {t}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t }^{2}}} \left( 2-\int \!8\,{\frac {t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}x}}\,{\rm d}t \right) }\,{\rm d}t}-{\frac {{ y}^{2}}{t-y}\int \!{\frac {8\,{t}^{3}{y}^{4}-2\,{t}^{3}+2\,{t}^{2}y+t{y }^{2}-{y}^{3}}{{t}^{2}}\int \!{\frac {t}{ \left( 8\,{t}^{3}{y}^{4}-2\,{ t}^{3}+2\,{t}^{2}y+t{y}^{2}-{y}^{3} \right) ^{2}} \left( {y}^{2} \left( 16\,{t}^{3}{y}^{3}+6\,{t}^{2}-6\,ty+{y}^{2} \right) +2\,{t}^{2} \left( {y}^{2}-5\,{t}^{2}-12\,{y}^{4}{t}^{2}+6\,{\frac {{t}^{3}}{y}}-2 \,{\frac {{t}^{4}}{{y}^{2}}}+24\,{y}^{2}{t}^{4} \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}- \left( 48\,{t}^{4}{y }^{4}-16\,{t}^{3}{y}^{5}-4\,{t}^{4}+16\,{t}^{3}y-16\,{t}^{2}{y}^{2}+6\, t{y}^{3}-{y}^{4} \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} \right) {\frac {1}{ \sqrt {1-2\,{\frac {t}{y}}- \left( 4\,{y}^{2}-{y}^{-2} \right) {t}^{2}} }}}\,{\rm d}t{\frac {1}{\sqrt {1-2\,{\frac {t}{y}}- \left( 4\,{y}^{2}-{ y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t} \right) } 06 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -4\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( t-1 \right) {t}^{2}}\int \!\!\!\int \!-2+\int \!6\,{ \frac {1-9\,t}{ \left( -15\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 1- \int \!4\,{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( - 4\,{t}^{2}+1 \right) ^{9/2} \left( 9\,t-1 \right) ^{2} \left( t+1 \right) } \left( \left( 1830\,{t}^{5}+59\,{t}^{4}+362\,{t}^{3}+22\,{t }^{2}-8\,t+3 \right) \left( -4\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +7\, \left( 750\,{t}^{4}-307\,{t}^{3}-213\,{t}^{2}-45\,t+11 \right) {t} ^{3} {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!-1/2+{\frac {1}{ \left( -4\,{t}^{2}+1 \right) ^{3/2}} \left( \left( 1/2+2\,t+6\,{t}^{2} \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -6\,{t}^{2} \left( t+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{t \left( t-1 \right) }\int \!-1+\int \!{\frac {t}{ \left( - 15\,{t}^{2}-2\,t+1 \right) ^{3/2} \left( 5\,t-1 \right) } \left( 26+ \int \!2\,{\frac { \left( 5\,t-1 \right) \sqrt {-15\,{t}^{2}-2\,t+1}}{{ t}^{2} \left( -4\,{t}^{2}+1 \right) ^{9/2}} \left( \left( 12\,{t}^{2}+ 4\,t+1 \right) \left( 240\,{t}^{5}+88\,{t}^{4}+64\,{t}^{3}+25\,{t}^{2} +2\,t+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -2\,{t}^{3} \left( 360\,{t}^{4}+128\,{t}^{3}+89\,{t}^{2}+52\,t+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1- \left( x+{x} ^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) }}{ \left( - 4\,{t}^{2}+1 \right) ^{7/2}{x}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( \left( {x}^{2}-4\,x+1 \right) t-x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -10\,{t}^{3} \left( t{x}^{2}+t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( t-y \right) {t}^{2}}\int \!\!\!\int \!-2\,y+\int \!{ (4\,t{y}^{4}+4\,t{y}^{3}+t-y) \left( -6\,{y}^{-1}+\int \!24\,{\frac {1 }{ \left( -4\,{t}^{2}+1 \right) ^{9/2}{y}^{2} \left( 4\,t{y}^{4}+4\,t{y }^{3}+t-y \right) ^{2} \left( t+1 \right) } \left( 1-2\,{\frac {t}{y}}- \left( \left( 2\,y+2 \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{ 3/2} \left( \left( \left( 576\,{y}^{7}+1408\,{y}^{6}+448\,{y}^{5}-608 \,{y}^{4}-72\,{y}^{3}+172\,{y}^{2}-60\,y-34 \right) {t}^{5}+ \left( 96 \,{y}^{7}+608\,{y}^{6}+128\,{y}^{5}-620\,{y}^{4}-264\,{y}^{3}+112\,{y}^ {2}-2\,y+1 \right) {t}^{4}+ \left( 96\,{y}^{7}+368\,{y}^{6}+124\,{y}^{5 }-174\,{y}^{4}-104\,{y}^{3}+63\,{y}^{2}-12\,y+1 \right) {t}^{3}+ \left( 16\,{y}^{7}+48\,{y}^{6}+24\,{y}^{5}-24\,{y}^{4}-59\,{y}^{3}+18 \,{y}^{2}-2\,y+1 \right) {t}^{2}+y \left( {y}^{4}-{y}^{3}-7\,{y}^{2}+y- 2 \right) t+{y}^{2} \left( {y}^{3}+{y}^{2}+1 \right) \right) \left( - 4\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +7\, \left( \left( 192\,{y}^{7}+576\,{y}^{6}+256\,{y}^{5}-256\,{y}^{4} -64\,{y}^{3}+84\,{y}^{2}-20\,y-18 \right) {t}^{4}+ \left( -96\,{y}^{7}+ 112\,{y}^{6}+72\,{y}^{5}-276\,{y}^{4}-172\,{y}^{3}+48\,{y}^{2}+6\,y-1 \right) {t}^{3}+ \left( -48\,{y}^{7}+16\,{y}^{6}-32\,{y}^{5}-102\,{y}^ {4}-62\,{y}^{3}+21\,{y}^{2}-8\,y+2 \right) {t}^{2}-y \left( 26\,{y}^{4} +20\,{y}^{3}+14\,{y}^{2}-19\,y+4 \right) t+{y}^{2} \left( 2\,{y}^{3}+2 \,{y}^{2}+7 \right) \right) {t}^{3} {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( \left( 2\,y+2 \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}y+{y}^{2}+x+y \right) } \left( xy-{\frac {x}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1- \left( x+{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) }}{ \left( - 4\,{t}^{2}+1 \right) ^{7/2}{x}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( \left( {x}^{2}-4\,x+1 \right) t-x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -10\,{t}^{3} \left( t{x}^{2}+t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac { \left( y+1 \right) y}{t \left( t-y \right) }\int \!\!\!\int \!-2\,y+\int \!{(4\,t {y}^{4}+4\,t{y}^{3}+t-y) \left( -6\,{y}^{-1}+\int \!24\,{\frac {1}{ \left( -4\,{t}^{2}+1 \right) ^{9/2}{y}^{2} \left( 4\,t{y}^{4}+4\,t{y}^ {3}+t-y \right) ^{2} \left( t+1 \right) } \left( 1-2\,{\frac {t}{y}}- \left( \left( 2\,y+2 \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{ 3/2} \left( \left( \left( 576\,{y}^{7}+1408\,{y}^{6}+448\,{y}^{5}-608 \,{y}^{4}-72\,{y}^{3}+172\,{y}^{2}-60\,y-34 \right) {t}^{5}+ \left( 96 \,{y}^{7}+608\,{y}^{6}+128\,{y}^{5}-620\,{y}^{4}-264\,{y}^{3}+112\,{y}^ {2}-2\,y+1 \right) {t}^{4}+ \left( 96\,{y}^{7}+368\,{y}^{6}+124\,{y}^{5 }-174\,{y}^{4}-104\,{y}^{3}+63\,{y}^{2}-12\,y+1 \right) {t}^{3}+ \left( 16\,{y}^{7}+48\,{y}^{6}+24\,{y}^{5}-24\,{y}^{4}-59\,{y}^{3}+18 \,{y}^{2}-2\,y+1 \right) {t}^{2}+y \left( {y}^{4}-{y}^{3}-7\,{y}^{2}+y- 2 \right) t+{y}^{2} \left( {y}^{3}+{y}^{2}+1 \right) \right) \left( - 4\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +7\, \left( \left( 192\,{y}^{7}+576\,{y}^{6}+256\,{y}^{5}-256\,{y}^{4} -64\,{y}^{3}+84\,{y}^{2}-20\,y-18 \right) {t}^{4}+ \left( -96\,{y}^{7}+ 112\,{y}^{6}+72\,{y}^{5}-276\,{y}^{4}-172\,{y}^{3}+48\,{y}^{2}+6\,y-1 \right) {t}^{3}+ \left( -48\,{y}^{7}+16\,{y}^{6}-32\,{y}^{5}-102\,{y}^ {4}-62\,{y}^{3}+21\,{y}^{2}-8\,y+2 \right) {t}^{2}-y \left( 26\,{y}^{4} +20\,{y}^{3}+14\,{y}^{2}-19\,y+4 \right) t+{y}^{2} \left( 2\,{y}^{3}+2 \,{y}^{2}+7 \right) \right) {t}^{3} {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( \left( 2\,y+2 \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t} \right) }$
07 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!2\,{\frac {t}{\sqrt {4\,{t}^{2}+1}} {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t} 01\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle {\frac {1}{t \left( t-1 \right) } \left( \int \!{\frac {4\,{t}^{2}+2\,t -1}{\sqrt {1-4\,t}{t}^{2}} \left( 1/4+\int \!{\frac {t}{\sqrt {1-4\,t} \sqrt {4\,{t}^{2}+1} \left( 4\,{t}^{2}+2\,t-1 \right) ^{2}} \left( \left( 1-4\,t \right) \left( 1-t \right) {{}_{2}F_{1}\!\left(1/2,3/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -t \left( 16\,{t}^{3}+24\,{t}^{2}-16\,t+1 \right) {{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t-1/4\,{t}^{-1}-t \right) }$
10$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!2\,{\frac {t}{\sqrt {4\,{t}^{2}+1}} {{}_{2}F_{1}\!\left(1/2,3/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle {\frac {1}{t \left( t-1 \right) }\int \!{\frac {t}{ \left( 1-4\,t \right) ^{3/2}} \left( 4+\int \!{\frac {\sqrt {1-4\,t} \left( t+1/2 \right) }{{t}^{2}} \left( 1+1/2\,{\frac {1}{\sqrt {4\,{t}^{2}+1}t \left( 2\,t+1 \right) } \left( \left( 1-t \right) {{}_{2}F_{1}\!\left(1/2,3/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} - \left( t+1 \right) \left( 8\,{t}^{2}-4\,t+1 \right) {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) }\,{\rm d}t \right) }\,{\rm d}t}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {1-4\, \left( x+1+{x}^{-1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t}{\sqrt {1-4\, \left( x+ 1+{x}^{-1} \right) {t}^{2}} \left( 4\,{t}^{2}+1 \right) ^{3/2}x} {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t \right) }\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle {\frac {1}{t \left( t-y \right) }\int \!{\frac {6\,{t}^{3}{y}^{4}-4\,{t }^{3}{y}^{2}+2\,{t}^{2}{y}^{3}+t{y}^{4}-2\,{t}^{3}+2\,{t}^{2}y+t{y}^{2} -{y}^{3}}{{t}^{2}}\int \!{\frac {t}{ \left( 6\,{t}^{3}{y}^{4}-4\,{t}^{3 }{y}^{2}+2\,{t}^{2}{y}^{3}+t{y}^{4}-2\,{t}^{3}+2\,{t}^{2}y+t{y}^{2}-{y} ^{3} \right) ^{2}} \left( {y}^{2} \left( 12\,{t}^{3}{y}^{3}-4\,{t}^{3}y +6\,{t}^{2}{y}^{2}+6\,{t}^{2}-6\,ty+{y}^{2} \right) +{\frac {1}{ \left( 4\,{t}^{2}+1 \right) ^{3/2}{y}^{2}} \left( 2\,{t}^{2} \left( 6 \,{t}^{4}{y}^{6}+6\,{t}^{3}{y}^{7}+18\,{t}^{4}{y}^{4}+2\,{t}^{3}{y}^{5} -7\,{t}^{2}{y}^{6}-6\,{y}^{2}{t}^{4}+10\,{t}^{3}{y}^{3}-4\,t{y}^{5}-2\, {t}^{4}+6\,{t}^{3}y-5\,{t}^{2}{y}^{2}+{y}^{4} \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -{y}^{2} \left( 4\,{t}^{2}+1 \right) \left( 36\,{t}^{4}{y}^{4}-12\,{t} ^{3}{y}^{5}-16\,{y}^{2}{t}^{4}+20\,{t}^{3}{y}^{3}-4\,{y}^{4}{t}^{2}-4\, {t}^{4}+16\,{t}^{3}y-16\,{t}^{2}{y}^{2}+6\,t{y}^{3}-{y}^{4} \right) {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) {\frac {1}{\sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}-2-{y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t{\frac {1}{ \sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}-2-{y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+x{y}^{2}+{y}^{2}+x \right) } \left( xy-{\frac {x}{t}\int \!{\frac {t}{\sqrt {1-4\, \left( x+1+{x}^{ -1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t}{\sqrt {1-4\, \left( x+1+{x}^{-1} \right) {t}^{2}} \left( 4\,{t}^{2}+1 \right) ^{3/2 }x} {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t \right) }\,{\rm d}t}-{\frac {{y}^{2}}{t-y}\int \!{\frac {6 \,{t}^{3}{y}^{4}-4\,{t}^{3}{y}^{2}+2\,{t}^{2}{y}^{3}+t{y}^{4}-2\,{t}^{3 }+2\,{t}^{2}y+t{y}^{2}-{y}^{3}}{{t}^{2}}\int \!{\frac {t}{ \left( 6\,{t }^{3}{y}^{4}-4\,{t}^{3}{y}^{2}+2\,{t}^{2}{y}^{3}+t{y}^{4}-2\,{t}^{3}+2 \,{t}^{2}y+t{y}^{2}-{y}^{3} \right) ^{2}} \left( {y}^{2} \left( 12\,{t} ^{3}{y}^{3}-4\,{t}^{3}y+6\,{t}^{2}{y}^{2}+6\,{t}^{2}-6\,ty+{y}^{2} \right) +{\frac {1}{ \left( 4\,{t}^{2}+1 \right) ^{3/2}{y}^{2}} \left( 2\,{t}^{2} \left( 6\,{t}^{4}{y}^{6}+6\,{t}^{3}{y}^{7}+18\,{t}^{ 4}{y}^{4}+2\,{t}^{3}{y}^{5}-7\,{t}^{2}{y}^{6}-6\,{y}^{2}{t}^{4}+10\,{t} ^{3}{y}^{3}-4\,t{y}^{5}-2\,{t}^{4}+6\,{t}^{3}y-5\,{t}^{2}{y}^{2}+{y}^{4 } \right) {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -{y}^{2} \left( 4\,{t}^{2}+1 \right) \left( 36\,{t}^{4}{y}^{4}-12\,{t} ^{3}{y}^{5}-16\,{y}^{2}{t}^{4}+20\,{t}^{3}{y}^{3}-4\,{y}^{4}{t}^{2}-4\, {t}^{4}+16\,{t}^{3}y-16\,{t}^{2}{y}^{2}+6\,t{y}^{3}-{y}^{4} \right) {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) {\frac {1}{\sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}-2-{y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t{\frac {1}{ \sqrt {1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}-2-{y}^{-2} \right) {t}^{2}}}}}\,{\rm d}t} \right) } 08 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -8\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle 1+6\,{\frac {1}{{t}^{2}}\int \!\!\!\int \!\!\!\int \!{\frac {t}{ \left( -12\,{t}^{2}-4\,t+1 \right) ^{5/2}} \left( 14+\int \!{\frac { \left( -12\,{t}^{2}-4\,t+1 \right) ^{3/2}}{ \left( 2\,t+1 \right) \left( -8\,{t}^{2}+1 \right) ^{7/2}{t}^{2}} \left( \left( -160\,{t}^{ 4}-176\,{t}^{3}-56\,{t}^{2}-8\,t-1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +4\,{t}^{2} \left( 32\,{t}^{3}-10\,{t}^{2}-19\,t-4 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -8\,{t}^{2}- 4\,t+1 \right) \left( -8\,{t}^{2}+1 \right) ^{3/2}} \left( \left( -8 \,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} -t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{t}\int \!1+\int \!{\frac { \left( 2\,t+1 \right) \left( 6\, t+1 \right) }{ \left( -12\,{t}^{2}-4\,t+1 \right) ^{5/2}} \left( 6+ \int \!3/2\,{\frac { \left( -12\,{t}^{2}-4\,t+1 \right) ^{3/2}}{ \left( 2\,t+1 \right) \left( 6\,t+1 \right) ^{2} \left( -8\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 8\,{t}^{2}-1 \right) \left( 96\,{t}^{3 }-248\,{t}^{2}-150\,t-13 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +5\, \left( 8\,{t}^{2}+4\,t+1 \right) \left( 32\,{t}^{3}-32\,{t}^{2}- 42\,t-5 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( x+ {x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2} \right) ^{3/2}} \left( 2-\int \!6\,{\frac {\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2}}}{{x}^{2} \left( -8\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 4\,{t}^{2}x+ \left( -{x}^{2}+4\,x-1 \right) t+x \right) \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\,{t}^{3} \left( \left( 1+x \right) ^{2}t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2} \left( ty-t+y \right) }\int \!\!\!\int \!2\,y+\int \!{(3\,t{y}^{3}+t{y}^{2}-ty+{y}^{2}+t-y) \left( {\frac {6\,{y}^{2}+6\,y -6}{y \left( y-1 \right) }}+\int \!-24\,{\frac {1}{ \left( -8\,{t}^{2}+ 1 \right) ^{9/2} \left( 3\,t{y}^{3}+t{y}^{2}-ty+{y}^{2}+t-y \right) ^{2 } \left( 2\,t+1 \right) {y}^{2}} \left( 1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}+8\,y+2-{y}^{-2} \right) {t}^{2} \right) ^ {3/2} \left( - \left( 8\,{t}^{2}-1 \right) \left( \left( 351\,{y}^{6} +720\,{y}^{5}-729\,{y}^{4}-496\,{y}^{3}+185\,{y}^{2}-128\,y-31 \right) {t}^{5}+ \left( 108\,{y}^{6}+804\,{y}^{5}-246\,{y}^{4}-692\,{y}^{3}+256 \,{y}^{2}-32\,y+2 \right) {t}^{4}+ \left( 63\,{y}^{6}+342\,{y}^{5}+104 \,{y}^{4}-373\,{y}^{3}+137\,{y}^{2}-23\,y+2 \right) {t}^{3}+ \left( 9\, {y}^{6}+72\,{y}^{5}+81\,{y}^{4}-125\,{y}^{3}+40\,{y}^{2}-6\,y+1 \right) {t}^{2}+2\,y \left( 3\,{y}^{4}+9\,{y}^{3}-10\,{y}^{2}+3\,y-1 \right) t+{y}^{2} \left( 2\,{y}^{2}-2\,y+1 \right) \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +7\, \left( \left( 252\,{y}^{6}+660\,{y}^{5}-688\,{y}^{4}-512\,{y}^{3} +180\,{y}^{2}-116\,y-32 \right) {t}^{4}-2\, \left( y+1 \right) \left( 9\,{y}^{5}-363\,{y}^{4}+516\,{y}^{3}-146\,{y}^{2}+15\,y+1 \right) {t}^{ 3}+ \left( -27\,{y}^{6}+243\,{y}^{5}+66\,{y}^{4}-392\,{y}^{3}+155\,{y}^ {2}-15\,y+2 \right) {t}^{2}+y \left( 27\,{y}^{4}+59\,{y}^{3}-113\,{y}^{ 2}+51\,y-4 \right) t+{y}^{2} \left( 9\,{y}^{2}-14\,y+7 \right) \right) {t}^{3} {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}+8\,y+2-{y}^{-2} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}y+x{y}^{2}+{y}^{2}+x+y \right) } \left( xy-{\frac {x}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2} \right) ^{3/2}} \left( 2-\int \!6\,{\frac { \sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2}}}{{x}^{2} \left( -8\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 4\,{t}^{2}x+ \left( -{x}^{2}+4\,x-1 \right) t+x \right) \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\,{t}^{3} \left( \left( 1+x \right) ^{2}t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac { \left( y+1 \right) y}{t \left( ty-t+y \right) }\int \!\!\!\int \!2\,y+\int \!{(3 \,t{y}^{3}+t{y}^{2}-ty+{y}^{2}+t-y) \left( {\frac {6\,{y}^{2}+6\,y-6}{y \left( y-1 \right) }}+\int \!-24\,{\frac {1}{ \left( -8\,{t}^{2}+1 \right) ^{9/2} \left( 3\,t{y}^{3}+t{y}^{2}-ty+{y}^{2}+t-y \right) ^{2} \left( 2\,t+1 \right) {y}^{2}} \left( 1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}+8\,y+2-{y}^{-2} \right) {t}^{2} \right) ^{3/2} \left( - \left( 8\,{t}^{2}-1 \right) \left( \left( 351\,{y}^{6}+720 \,{y}^{5}-729\,{y}^{4}-496\,{y}^{3}+185\,{y}^{2}-128\,y-31 \right) {t}^ {5}+ \left( 108\,{y}^{6}+804\,{y}^{5}-246\,{y}^{4}-692\,{y}^{3}+256\,{y }^{2}-32\,y+2 \right) {t}^{4}+ \left( 63\,{y}^{6}+342\,{y}^{5}+104\,{y} ^{4}-373\,{y}^{3}+137\,{y}^{2}-23\,y+2 \right) {t}^{3}+ \left( 9\,{y}^{ 6}+72\,{y}^{5}+81\,{y}^{4}-125\,{y}^{3}+40\,{y}^{2}-6\,y+1 \right) {t}^ {2}+2\,y \left( 3\,{y}^{4}+9\,{y}^{3}-10\,{y}^{2}+3\,y-1 \right) t+{y}^ {2} \left( 2\,{y}^{2}-2\,y+1 \right) \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +7\, \left( \left( 252\,{y}^{6}+660\,{y}^{5}-688\,{y}^{4}-512\,{y}^{3} +180\,{y}^{2}-116\,y-32 \right) {t}^{4}-2\, \left( y+1 \right) \left( 9\,{y}^{5}-363\,{y}^{4}+516\,{y}^{3}-146\,{y}^{2}+15\,y+1 \right) {t}^{ 3}+ \left( -27\,{y}^{6}+243\,{y}^{5}+66\,{y}^{4}-392\,{y}^{3}+155\,{y}^ {2}-15\,y+2 \right) {t}^{2}+y \left( 27\,{y}^{4}+59\,{y}^{3}-113\,{y}^{ 2}+51\,y-4 \right) t+{y}^{2} \left( 9\,{y}^{2}-14\,y+7 \right) \right) {t}^{3} {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\, \left( y+{y}^{-1} \right) t- \left( 3\,{y}^{2}+8\,y+2-{y}^{-2} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t} \right) }$
09 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 16\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} 01\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2} \left( t+1 \right) }\int \!\!\!\int \!2+\int \!12\,{ \frac {1-t}{ \left( -15\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 1-2\, \int \!{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 16\, {t}^{2}+1 \right) ^{11/2} \left( 1-t \right) ^{2}} \left( \left( 16\,{ t}^{2}+1 \right) \left( 4680\,{t}^{6}-1400\,{t}^{5}-185\,{t}^{4}-521\, {t}^{3}+123\,{t}^{2}+9\,t-2 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( -180\,{t}^{6}+644\,{t}^{5}+458\,{t}^{4}-1622\,{t}^{3}+356\, {t}^{2}-26\,t-2 \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{5/2}} \left( 2\, \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} +12\,{t}^{2} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{ \left( t+1 \right) t}\int \!1+\int \!{\frac { \left( t+1 \right) \left( 3\,t+1 \right) }{ \left( -15\,{t}^{2}-2\,t+1 \right) ^ {5/2}} \left( 6+\int \!6\,{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^ {3/2}}{ \left( 3\,t+1 \right) \left( 16\,{t}^{2}+1 \right) ^{9/2} \left( t+1 \right) ^{2}} \left( 4\,t \left( 360\,{t}^{6}+397\,{t}^{5}+ 504\,{t}^{4}-76\,{t}^{3}+89\,{t}^{2}+7\,t+5 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 8\,{t}^{2}-1 \right) \left( 4320\,{t}^{5}+1356\,{t}^{4}+112\, {t}^{3}-183\,{t}^{2}-38\,t+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+ {x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2} \right) ^{3/2}} \left( 2+24\,\int \!{\frac {\sqrt {1-4\, \left( x+{x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2}}t}{ \left( 16\,{t}^{2}+1 \right) ^ {7/2}{x}^{2}} \left( 10\, \left( 2\,{x}^{2}+x+2 \right) {t}^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( {x}^{2}+x+1 \right) \left( 16\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{ \left( t+y \right) {t}^{2}}\int \!\!\!\int \!2\,y+\int \!{( 3\,t{y}^{4}+{y}^{3}-4\,t) \left( 6\,{\frac {{y}^{2}+1}{{y}^{3}}}+\int \!-24\,{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{11/2} \left( 3\,t{y}^ {4}+{y}^{3}-4\,t \right) ^{2}{y}^{2}} \left( 1-2\,ty- \left( 3\,y+2\,{y }^{-1} \right) \left( y+2\,{y}^{-1} \right) {t}^{2} \right) ^{3/2} \left( \left( 16\,{t}^{2}+1 \right) \left( 40\, \left( 7\,{y}^{2}+6 \right) \left( 6\,{y}^{4}+5\,{y}^{2}-2 \right) {t}^{6}+40\,y \left( 18\,{y}^{6}-6\,{y}^{4}-39\,{y}^{2}-8 \right) {t}^{5}+ \left( -1257\,{y} ^{6}-944\,{y}^{4}+1392\,{y}^{2}+624 \right) {t}^{4}+y \left( 45\,{y}^{6 }-218\,{y}^{4}-188\,{y}^{2}-160 \right) {t}^{3}+ \left( 33\,{y}^{6}+90 \,{y}^{4}+16\,{y}^{2}-16 \right) {t}^{2}+{y}^{3} \left( 7\,{y}^{2}+2 \right) t-{y}^{2} \left( {y}^{2}+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 588\,{y}^{6}+496\,{y}^{4}-688\,{y}^{2}-576 \right) {t}^{6}-4\,y \left( 225\,{y}^{6}+282\,{y}^{4}-288\,{y}^{2}-380 \right) {t}^{5}+ \left( -2004\,{y}^{6}-1554\,{y}^{4}+2368\,{y}^{2}+1648 \right) {t}^{4}+2\,y \left( 180\,{y}^{6}-88\,{y}^{4}-543\,{y}^{2}-360 \right) {t}^{3}+ \left( -72\,{y}^{6}+189\,{y}^{4}+255\,{y}^{2}-16 \right) {t}^{2}-2\,{y}^{3} \left( 14\,{y}^{2}-1 \right) t-{y}^{2} \left( {y}^{2}+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,ty- \left( 3\,y+2\,{y}^{-1} \right) \left( y+2\,{y}^{-1} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+x{y}^{2}+{x}^{2}+{y}^{2}+1 \right) } \left( xy-{\frac {{x}^{2}+1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+{x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t }^{2} \right) ^{3/2}} \left( 2+24\,\int \!{\frac {\sqrt {1-4\, \left( x +{x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2}}t}{ \left( 16\,{ t}^{2}+1 \right) ^{7/2}{x}^{2}} \left( 10\, \left( 2\,{x}^{2}+x+2 \right) {t}^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( {x}^{2}+x+1 \right) \left( 16\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+1}{ t \left( t+y \right) }\int \!\!\!\int \!2\,y+\int \!{(3\,t{y}^{4}+{y}^{ 3}-4\,t) \left( 6\,{\frac {{y}^{2}+1}{{y}^{3}}}+\int \!-24\,{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{11/2} \left( 3\,t{y}^{4}+{y}^{3}-4\,t \right) ^{2}{y}^{2}} \left( 1-2\,ty- \left( 3\,y+2\,{y}^{-1} \right) \left( y+2\,{y}^{-1} \right) {t}^{2} \right) ^{3/2} \left( \left( 16 \,{t}^{2}+1 \right) \left( 40\, \left( 7\,{y}^{2}+6 \right) \left( 6 \,{y}^{4}+5\,{y}^{2}-2 \right) {t}^{6}+40\,y \left( 18\,{y}^{6}-6\,{y}^ {4}-39\,{y}^{2}-8 \right) {t}^{5}+ \left( -1257\,{y}^{6}-944\,{y}^{4}+ 1392\,{y}^{2}+624 \right) {t}^{4}+y \left( 45\,{y}^{6}-218\,{y}^{4}-188 \,{y}^{2}-160 \right) {t}^{3}+ \left( 33\,{y}^{6}+90\,{y}^{4}+16\,{y}^{ 2}-16 \right) {t}^{2}+{y}^{3} \left( 7\,{y}^{2}+2 \right) t-{y}^{2} \left( {y}^{2}+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 588\,{y}^{6}+496\,{y}^{4}-688\,{y}^{2}-576 \right) {t}^{6}-4\,y \left( 225\,{y}^{6}+282\,{y}^{4}-288\,{y}^{2}-380 \right) {t}^{5}+ \left( -2004\,{y}^{6}-1554\,{y}^{4}+2368\,{y}^{2}+1648 \right) {t}^{4}+2\,y \left( 180\,{y}^{6}-88\,{y}^{4}-543\,{y}^{2}-360 \right) {t}^{3}+ \left( -72\,{y}^{6}+189\,{y}^{4}+255\,{y}^{2}-16 \right) {t}^{2}-2\,{y}^{3} \left( 14\,{y}^{2}-1 \right) t-{y}^{2} \left( {y}^{2}+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,ty- \left( 3\,y+2\,{y}^{-1} \right) \left( y+2\,{y}^{-1} \right) {t}^{2} \right) ^{-5/2}} \,{\rm d}t\,{\rm d}t\,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{ \frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) } 10 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2} \left( 2\,t+1 \right) }\int \!\!\!\int \!2+\int \!6 \,{\frac {1-t}{ \left( -35\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 3- \int \!4\,{\frac { \left( -35\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 12\,{t}^{2}+1 \right) ^{11/2} \left( 1-t \right) ^{2}} \left( \left( 12\,{t}^{2}+1 \right) \left( -2520\,{t}^{6}+420\,{t}^{5}+930\,{t}^{4}- 225\,{t}^{3}+385\,{t}^{2}+25\,t-5 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} +6\, \left( 280\,{t}^{6}-980\,{t}^{5}+1830\,{t}^{4}+1875\,{t}^{3}-640\, {t}^{2}+15\,t+5 \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -20\,{t}^{2} -4\,t+1 \right) \left( 12\,{t}^{2}+1 \right) ^{3/2}} \left( \left( 12 \,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -t \left( 8\,t+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( 2\,t+1 \right) t}\int \!1+\int \!{\frac { \left( 2\, t+1 \right) \left( 5\,t+1 \right) }{ \left( -35\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 10+\int \!12\,{\frac { \left( -35\,{t}^{2}-2\,t +1 \right) ^{3/2}}{ \left( 5\,t+1 \right) \left( 12\,{t}^{2}+1 \right) ^{9/2} \left( 2\,t+1 \right) ^{2}} \left( \left( 12\,{t}^{2}+ 1 \right) \left( 736\,{t}^{5}+2208\,{t}^{4}+1096\,{t}^{3}-44\,{t}^{2}+ 44\,t+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -7\,t \left( 1824\,{t}^{6}+2496\,{t}^{5}+1288\,{t}^{4}+452\,{t}^{3}+420 \,{t}^{2}+53\,t+10 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{1 \left( 2+\int \!6\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{x}^{2}}\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) } \left( 30\,{t}^{3} \left( t{x}^{2}+2\,{x}^{2}+t+x+2 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( \left( 3\,{x}^{2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) \right) ^{-3/2} }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2} \left( ty+t+y \right) }\int \!\!\!\int \!2\,y+\int \!{\frac { \left( 3\,{y}^{4}+4\,{y}^{3}-4\,y-4 \right) t+{y}^{3}}{ \left( 1-2\,ty- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{5/2}} \left( 6\,{\frac {{y}^{2}+y+1}{{y}^{3} }}+\int \!24\,{\frac { \left( 1-2\,ty- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{3/2}}{ \left( 12\,{t}^{ 2}+1 \right) ^{11/2}{y}^{2} \left( \left( 3\,{y}^{4}+4\,{y}^{3}-4\,y-4 \right) t+{y}^{3} \right) ^{2}} \left( - \left( 12\,{t}^{2}+1 \right) \left( \left( 1620\,{y}^{7}+3708\,{y}^{6}+2256\,{y}^{5}-456\,{y}^{4}- 3856\,{y}^{3}-2912\,{y}^{2}-2272\,y-608 \right) {t}^{6}+ \left( 1080\,{ y}^{7}+864\,{y}^{6}-480\,{y}^{5}-1476\,{y}^{4}-552\,{y}^{3}+504\,{y}^{2 }+144\,y+336 \right) {t}^{5}+ \left( 135\,{y}^{7}-117\,{y}^{6}-1128\,{y }^{5}-1296\,{y}^{4}-24\,{y}^{3}+1200\,{y}^{2}+1488\,y+672 \right) {t}^{ 4}+y \left( 90\,{y}^{6}+555\,{y}^{5}+632\,{y}^{4}+104\,{y}^{3}-652\,{y} ^{2}-714\,y-240 \right) {t}^{3}+ \left( 66\,{y}^{6}+217\,{y}^{5}+192\,{ y}^{4}+42\,{y}^{3}-52\,{y}^{2}-64\,y-16 \right) {t}^{2}+{y}^{3} \left( 14\,{y}^{2}+9\,y+2 \right) t-{y}^{2} \left( 2\,{y}^{2}+2\,y+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 180\,{y}^{7}+468\,{y}^{6}+736\,{y}^{5}+304\,{y}^{4} -416\,{y}^{3}-672\,{y}^{2}-352\,y+32 \right) {t}^{6}+ \left( 720\,{y}^{ 7}+3588\,{y}^{6}+5928\,{y}^{5}+3696\,{y}^{4}-2400\,{y}^{3}-6272\,{y}^{2 }-4592\,y-1648 \right) {t}^{5}+ \left( -225\,{y}^{7}+1737\,{y}^{6}+4276 \,{y}^{5}+3802\,{y}^{4}+236\,{y}^{3}-3516\,{y}^{2}-2848\,y-1632 \right) {t}^{4}+ \left( -405\,{y}^{7}-621\,{y}^{6}-114\,{y}^{5}+129\,{ y}^{4}+1070\,{y}^{3}+936\,{y}^{2}+864\,y+16 \right) {t}^{3}+ \left( 39 \,{y}^{6}+49\,{y}^{5}-321\,{y}^{4}-304\,{y}^{3}-183\,{y}^{2}+64\,y+16 \right) {t}^{2}+{y}^{2} \left( 21\,{y}^{3}-7\,{y}^{2}+1 \right) t+{y}^ {2} \left( 2\,{y}^{2}+2\,y+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}y+x{y}^{2}+{x}^{2}+{y}^{2} +y+1 \right) } \left( xy-{\frac {{x}^{2}+1}{t}\int \!\!\!\int \!{1 \left( 2+\int \!6\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{x}^ {2}}\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x }^{-1} \right) t \right) } \left( 30\,{t}^{3} \left( t{x}^{2}+2\,{x}^{2 }+t+x+2 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( \left( 3\,{x}^{2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) \right) ^{-3/2} }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+y+1}{t \left( ty+t+y \right) } \int \!\!\!\int \!2\,y+\int \!{\frac { \left( 3\,{y}^{4}+4\,{y}^{3}-4\, y-4 \right) t+{y}^{3}}{ \left( 1-2\,ty- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{5/2}} \left( 6\,{\frac {{y}^{2}+y+1}{{y}^{3}}}+\int \!24\,{\frac { \left( 1-2\,ty- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{3/2 }}{ \left( 12\,{t}^{2}+1 \right) ^{11/2}{y}^{2} \left( \left( 3\,{y}^{ 4}+4\,{y}^{3}-4\,y-4 \right) t+{y}^{3} \right) ^{2}} \left( - \left( 12 \,{t}^{2}+1 \right) \left( \left( 1620\,{y}^{7}+3708\,{y}^{6}+2256\,{ y}^{5}-456\,{y}^{4}-3856\,{y}^{3}-2912\,{y}^{2}-2272\,y-608 \right) {t} ^{6}+ \left( 1080\,{y}^{7}+864\,{y}^{6}-480\,{y}^{5}-1476\,{y}^{4}-552 \,{y}^{3}+504\,{y}^{2}+144\,y+336 \right) {t}^{5}+ \left( 135\,{y}^{7}- 117\,{y}^{6}-1128\,{y}^{5}-1296\,{y}^{4}-24\,{y}^{3}+1200\,{y}^{2}+1488 \,y+672 \right) {t}^{4}+y \left( 90\,{y}^{6}+555\,{y}^{5}+632\,{y}^{4}+ 104\,{y}^{3}-652\,{y}^{2}-714\,y-240 \right) {t}^{3}+ \left( 66\,{y}^{6 }+217\,{y}^{5}+192\,{y}^{4}+42\,{y}^{3}-52\,{y}^{2}-64\,y-16 \right) {t }^{2}+{y}^{3} \left( 14\,{y}^{2}+9\,y+2 \right) t-{y}^{2} \left( 2\,{y} ^{2}+2\,y+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 180\,{y}^{7}+468\,{y}^{6}+736\,{y}^{5}+304\,{y}^{4} -416\,{y}^{3}-672\,{y}^{2}-352\,y+32 \right) {t}^{6}+ \left( 720\,{y}^{ 7}+3588\,{y}^{6}+5928\,{y}^{5}+3696\,{y}^{4}-2400\,{y}^{3}-6272\,{y}^{2 }-4592\,y-1648 \right) {t}^{5}+ \left( -225\,{y}^{7}+1737\,{y}^{6}+4276 \,{y}^{5}+3802\,{y}^{4}+236\,{y}^{3}-3516\,{y}^{2}-2848\,y-1632 \right) {t}^{4}+ \left( -405\,{y}^{7}-621\,{y}^{6}-114\,{y}^{5}+129\,{ y}^{4}+1070\,{y}^{3}+936\,{y}^{2}+864\,y+16 \right) {t}^{3}+ \left( 39 \,{y}^{6}+49\,{y}^{5}-321\,{y}^{4}-304\,{y}^{3}-183\,{y}^{2}+64\,y+16 \right) {t}^{2}+{y}^{2} \left( 21\,{y}^{3}-7\,{y}^{2}+1 \right) t+{y}^ {2} \left( 2\,{y}^{2}+2\,y+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t}+{\frac {1 }{t}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{3/2 }} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) }$
11 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!2\,{\frac {t}{\sqrt {4\,{t}^{2}+1}} {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t} 01\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle {\frac {\sqrt {1-4\,t} \left( 2\,t-1 \right) }{{t}^{3}}\int \!{\frac {t }{ \left( 1-4\,t \right) ^{3/2}} \left( 1+{\frac {1}{\sqrt {4\,{t}^{2}+ 1} \left( 2\,t-1 \right) ^{2}} \left( \left( 3\,t-1 \right) \left( 2 \,t+1 \right) {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -2\,{t}^{2} \left( 2\,t-1 \right) {{}_{2}F_{1}\!\left(3/2,3/2;4;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) }\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!2\,{\frac {t}{\sqrt {4\,{t}^{2}+1}} {{}_{2}F_{1}\!\left(1/2,3/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle 1/4\,{\frac {1-2\,t}{{t}^{2}} \left( 1+{\frac {1}{\sqrt {1-4\,t}} \left( -1+2\,\int \!{\frac {1}{\sqrt {1-4\,t} \left( 1-2\,t \right) ^{ 2}\sqrt {4\,{t}^{2}+1}} \left( \left( -24\,{t}^{3}+1 \right) {{}_{2}F_{1}\!\left(1/2,3/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} +18\,{t}^{2} \left( 2\,t-1 \right) {{}_{2}F_{1}\!\left(1/2,5/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) }\,{\rm d}t \right) } \right) }$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {1-4\, \left( x+1+{x}^{-1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t}{\sqrt {1-4\, \left( x+ 1+{x}^{-1} \right) {t}^{2}} \left( 4\,{t}^{2}+1 \right) ^{3/2}x} {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t \right) }\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)}$$\displaystyle {\frac {1}{{t}^{3}}\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( y+3 \,{y}^{-1} \right) \left( y-{y}^{-1} \right) {t}^{2}} \left( 2\,t-{y}^ {-1} \right) \int \!{t \left( 1+{\frac {1}{\sqrt {4\,{t}^{2}+1} \left( 2\,ty-1 \right) ^{2}y} \left( y \left( 4\,{t}^{2}+1 \right) \left( 2\, ty-1 \right) {{}_{2}F_{1}\!\left(-1/2,1/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -t \left( t{y}^{2}-t-y \right) \left( 6\,t-y \right) {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y+ 3\,{y}^{-1} \right) \left( y-{y}^{-1} \right) {t}^{2} \right) ^{-3/2}} \,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} $$\displaystyle {\frac {1}{xy-t \left( x{y}^{2}+{x}^{2}+x+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!{\frac {t}{\sqrt {1-4\, \left( x+1+{x}^{-1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t}{\sqrt {1-4\, \left( x+ 1+{x}^{-1} \right) {t}^{2}} \left( 4\,{t}^{2}+1 \right) ^{3/2}x} {{}_{2}F_{1}\!\left(3/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t \right) }\,{\rm d}t}-{\frac {1}{{t}^{2}}\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( y+3\,{y}^{-1} \right) \left( y-{y }^{-1} \right) {t}^{2}} \left( 2\,t-{y}^{-1} \right) \int \!{t \left( 1 +{\frac {1}{\sqrt {4\,{t}^{2}+1} \left( 2\,ty-1 \right) ^{2}y} \left( y \left( 4\,{t}^{2}+1 \right) \left( 2\,ty-1 \right) {{}_{2}F_{1}\!\left(-1/2,1/2;2;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} -t \left( t{y}^{2}-t-y \right) \left( 6\,t-y \right) {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} \right) } \right) \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y+ 3\,{y}^{-1} \right) \left( y-{y}^{-1} \right) {t}^{2} \right) ^{-3/2}} \,{\rm d}t}+{\frac {1}{t}\int \!2\,{\frac {t}{\sqrt {4\,{t}^{2}+1}} {{}_{2}F_{1}\!\left(1/2,3/2;3;16\,{\frac {{t}^{2}}{4\,{t}^{2}+1}}\right)} }\,{\rm d}t} \right) } 12 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -8\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{3}}\int \!\!\!\int \!\!\!\int \!{\frac {t}{ \left( -12 \,{t}^{2}-4\,t+1 \right) ^{5/2}} \left( -36-2\,\int \!{\frac { \left( - 12\,{t}^{2}-4\,t+1 \right) ^{3/2}}{ \left( 2\,t+1 \right) \left( -8\,{ t}^{2}+1 \right) ^{7/2}{t}^{2}} \left( \left( 256\,{t}^{5}+416\,{t}^{4 }+128\,{t}^{3}+3\,t+3 \right) {{}_{2}F_{1}\!\left(5/4,7/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} -7\,t \left( 40\,{t}^{4}+68\,{t}^{3}-4\,{t}^{2}-18\,t-3 \right) {{}_{2}F_{1}\!\left(5/4,11/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -8\,{t}^{2}- 4\,t+1 \right) \left( -8\,{t}^{2}+1 \right) ^{3/2}} \left( \left( -8 \,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} -t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac { \left( 2\,t+1 \right) \left( 6\,t+1 \right) }{ \left( -12\,{t}^{2}-4\,t+1 \right) ^{5/2}} \left( 2-\int \!6\,{\frac { \left( -12\,{t}^{2}-4\,t+1 \right) ^{3/2} }{ \left( 2\,t+1 \right) \left( 6\,t+1 \right) ^{2} \left( -8\,{t}^{2} +1 \right) ^{7/2}} \left( \left( 10\,{t}^{2}+17\,t+4 \right) \left( 8 \,{t}^{2}+4\,t+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} -t \left( 304\,{t}^{4}+96\,{t}^{3}-124\,{t}^{2}-50\,t-3 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( x+ {x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2} \right) ^{3/2}} \left( 2-\int \!6\,{\frac {\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2}}}{{x}^{2} \left( -8\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 4\,{t}^{2}x+ \left( -{x}^{2}+4\,x-1 \right) t+x \right) \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\,{t}^{3} \left( \left( 1+x \right) ^{2}t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} $$\displaystyle {\frac {ty-t-1}{{t}^{2}}\int \!{\frac { \left( 6\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{{t}^{2} \left( ty-t-1 \right) ^{2}}\int \!{\frac { \left( ty-t-1 \right) \left( \left( 3\, {y}^{2}+2\,y+1 \right) \left( {y}^{3}-3\,{y}^{2}-3\,y-3 \right) {t}^{3 }-3\,y \left( y-1 \right) \left( 3\,{y}^{2}+2\,y+1 \right) {t}^{2}+6\, t{y}^{3}-{y}^{2} \right) t}{ \left( \left( 6\,{y}^{2}+4\,y+2 \right) { t}^{2}+2\,y \left( y-1 \right) t-y \right) ^{2}}\int \!6\,{\frac { \left( 6\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{ \left( -8\,{t}^{2}+1 \right) ^{7/2}{y}^{2} \left( \left( 3\,{y}^{2}+2 \,y+1 \right) \left( {y}^{3}-3\,{y}^{2}-3\,y-3 \right) {t}^{3}-3\,y \left( y-1 \right) \left( 3\,{y}^{2}+2\,y+1 \right) {t}^{2}+6\,t{y}^{ 3}-{y}^{2} \right) ^{2}}\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( y+2+3\,{y}^{-1} \right) \left( y-2-{y}^{-1} \right) {t}^{2}} \left( - \left( 8\,{t}^{2}-1 \right) \left( 4\,y \left( 3\,{y}^{6}-16 \,{y}^{5}+11\,{y}^{4}+56\,{y}^{3}-83\,{y}^{2}-56\,y+21 \right) {t}^{6}+ \left( 3\,{y}^{7}-67\,{y}^{6}+105\,{y}^{5}+107\,{y}^{4}-695\,{y}^{3}- 377\,{y}^{2}+51\,y+9 \right) {t}^{5}-y \left( 15\,{y}^{5}-51\,{y}^{4}+ 42\,{y}^{3}+374\,{y}^{2}+155\,y+9 \right) {t}^{4}+2\,{y}^{2} \left( 2\, {y}^{3}-9\,{y}^{2}-22\,y-1 \right) {t}^{3}+2\,{y}^{2} \left( y+6 \right) \left( 2\,y+1 \right) {t}^{2}+{y}^{3} \left( {y}^{2}+5 \right) t-{y}^{4} \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\, \left( \left( 3\,{y}^{7}-{y}^{6}+45\,{y}^{5}+109\,{y}^{4}+25\,{y }^{3}+37\,{y}^{2}+15\,y-9 \right) {t}^{5}+y \left( 3\,{y}^{6}-10\,{y}^{ 5}+37\,{y}^{4}+88\,{y}^{3}-51\,{y}^{2}+18\,y+27 \right) {t}^{4}-{y}^{2} \left( 9\,{y}^{4}-21\,{y}^{3}-35\,{y}^{2}+79\,y+28 \right) {t}^{3}+{y} ^{2} \left( 7\,{y}^{3}+2\,{y}^{2}-41\,y-12 \right) {t}^{2}-{y}^{3} \left( {y}^{2}-y+5 \right) t+{y}^{4} \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y +2+3\,{y}^{-1} \right) \left( y-2-{y}^{-1} \right) {t}^{2} \right) ^{- 3/2}}\,{\rm d}t}\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}y+x{y}^{2}+{x}^{2}+x+y+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2} \right) ^{3/2}} \left( 2-\int \!6\,{\frac { \sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-8 \right) {t}^{2}}}{{x}^{2} \left( -8\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 4\,{t}^{2}x+ \left( -{x}^{2}+4\,x-1 \right) t+x \right) \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\,{t}^{3} \left( \left( 1+x \right) ^{2}t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac { \left( y+1 \right) \left( ty-t-1 \right) }{t}\int \!{\frac { \left( 6\,{y}^{2}+4 \,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{{t}^{2} \left( ty-t- 1 \right) ^{2}}\int \!{\frac { \left( ty-t-1 \right) \left( \left( 3 \,{y}^{2}+2\,y+1 \right) \left( {y}^{3}-3\,{y}^{2}-3\,y-3 \right) {t}^ {3}-3\,y \left( y-1 \right) \left( 3\,{y}^{2}+2\,y+1 \right) {t}^{2}+6 \,t{y}^{3}-{y}^{2} \right) t}{ \left( \left( 6\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y \right) ^{2}}\int \!6\,{ \frac { \left( 6\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{ \left( -8\,{t}^{2}+1 \right) ^{7/2}{y}^{2} \left( \left( 3\,{y}^{2}+2\,y+1 \right) \left( {y}^{3}-3\,{y}^{2}-3\,y-3 \right) {t}^{3}-3\,y \left( y-1 \right) \left( 3\,{y}^{2}+2\,y+1 \right) {t}^{2}+6\,t{y}^{3}-{y}^{2} \right) ^{2}}\sqrt {1-2\, \left( y +{y}^{-1} \right) t+ \left( y+2+3\,{y}^{-1} \right) \left( y-2-{y}^{-1 } \right) {t}^{2}} \left( - \left( 8\,{t}^{2}-1 \right) \left( 4\,y \left( 3\,{y}^{6}-16\,{y}^{5}+11\,{y}^{4}+56\,{y}^{3}-83\,{y}^{2}-56\, y+21 \right) {t}^{6}+ \left( 3\,{y}^{7}-67\,{y}^{6}+105\,{y}^{5}+107\,{ y}^{4}-695\,{y}^{3}-377\,{y}^{2}+51\,y+9 \right) {t}^{5}-y \left( 15\,{ y}^{5}-51\,{y}^{4}+42\,{y}^{3}+374\,{y}^{2}+155\,y+9 \right) {t}^{4}+2 \,{y}^{2} \left( 2\,{y}^{3}-9\,{y}^{2}-22\,y-1 \right) {t}^{3}+2\,{y}^{ 2} \left( y+6 \right) \left( 2\,y+1 \right) {t}^{2}+{y}^{3} \left( {y} ^{2}+5 \right) t-{y}^{4} \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} +10\, \left( \left( 3\,{y}^{7}-{y}^{6}+45\,{y}^{5}+109\,{y}^{4}+25\,{y }^{3}+37\,{y}^{2}+15\,y-9 \right) {t}^{5}+y \left( 3\,{y}^{6}-10\,{y}^{ 5}+37\,{y}^{4}+88\,{y}^{3}-51\,{y}^{2}+18\,y+27 \right) {t}^{4}-{y}^{2} \left( 9\,{y}^{4}-21\,{y}^{3}-35\,{y}^{2}+79\,y+28 \right) {t}^{3}+{y} ^{2} \left( 7\,{y}^{3}+2\,{y}^{2}-41\,y-12 \right) {t}^{2}-{y}^{3} \left( {y}^{2}-y+5 \right) t+{y}^{4} \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( y +2+3\,{y}^{-1} \right) \left( y-2-{y}^{-1} \right) {t}^{2} \right) ^{- 3/2}}\,{\rm d}t}\,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{\frac {1 }{ \left( -8\,{t}^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( 2\,t+1 \right) {t}^{3}}{ \left( 8\,{t}^{2}-1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) }$
13 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 16\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} 01\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{ \left( t-1 \right) {t}^{2}}\int \!\!\!\int \!{\frac {450\,{ t}^{3}+90\,{t}^{2}-42\,t-2}{ \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}} \left( 1+3\,\int \!{\frac {\sqrt {-15\,{t}^{2}-2\,t+1}}{ \left( 225\,{ t}^{3}+45\,{t}^{2}-21\,t-1 \right) ^{2}} \left( t-1+{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{7/2}} \left( \left( 16\,{t}^{2}+1 \right) \left( 19872\,{t}^{6}+3753\,{t}^{5}-5565\,{t}^{4}-510\,{t}^{3}+294\,{t }^{2}+20\,t-8 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -10\,{t}^{2} \left( 2880\,{t}^{6}-2205\,{t}^{5}-2139\,{t}^{4}+423\,{t}^ {3}+45\,{t}^{2}+12\,t-8 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) } \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{5/2}} \left( 2\, \left( -8\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} +12\,{t}^{2} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{ \left( t-1 \right) t}\int \!-1+\int \!{\frac { \left( t+1 \right) \left( 3\,t+1 \right) }{ \left( -15\,{t}^{2}-2\,t+1 \right) ^ {5/2}}\int \!12\,{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 3\,t+1 \right) \left( 16\,{t}^{2}+1 \right) ^{9/2} \left( t+1 \right) ^{2}} \left( \left( 8\,{t}^{2}-1 \right) \left( 2280\,{t}^{5 }+621\,{t}^{4}-104\,{t}^{3}-117\,{t}^{2}-14\,t+2 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -t \left( 280\,{t}^{6}-297\,{t}^{5}-562\,{t}^{4}-406\,{t}^{3}+128\,{t}^ {2}+11\,t+10 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t}\,{\rm d}t\,{\rm d}t}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+ {x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2} \right) ^{3/2}} \left( 2+24\,\int \!{\frac {\sqrt {1-4\, \left( x+{x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2}}t}{ \left( 16\,{t}^{2}+1 \right) ^ {7/2}{x}^{2}} \left( 10\, \left( 2\,{x}^{2}+x+2 \right) {t}^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( {x}^{2}+x+1 \right) \left( 16\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{ \left( t-y \right) {t}^{2}}\int \!\!\!\int \!-2\,y+\int \!{ (4\,t{y}^{4}-3\,t-y) \left( -6\,{y}^{-1}+\int \!-24\,{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{11/2}{y}^{2} \left( 4\,t{y}^{4}-3\,t-y \right) ^{2}} \left( 1-2\,{\frac {t}{y}}- \left( 2\,y+{y}^{-1} \right) \left( 2\,y+3\,{y}^{-1} \right) {t}^{2} \right) ^{3/2} \left( \left( 16\,{t}^{2}+1 \right) \left( 10\, \left( 10\,{y}^{2}+9 \right) \left( 16\,{y}^{4}+14\,{y}^{2}-3 \right) {t}^{6}-10\,y \left( 4\,{y}^{2}+3 \right) \left( 8\,{y}^{4}+12\,{y}^{2}-3 \right) { t}^{5}+ \left( -576\,{y}^{6}-944\,{y}^{4}+69\,{y}^{2}+351 \right) {t}^{ 4}-4\,y \left( 40\,{y}^{6}+20\,{y}^{4}-83\,{y}^{2}-66 \right) {t}^{3}+ \left( -16\,{y}^{6}+90\,{y}^{4}+102\,{y}^{2}-9 \right) {t}^{2}+2\,y \left( {y}^{2}-3 \right) t-{y}^{2} \left( {y}^{2}+2 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 464\,{y}^{6}+496\,{y}^{4}-276\,{y}^{2}-324 \right) {t}^{6}+8\,y \left( 190\,{y}^{6}+270\,{y}^{4}-141\,{y}^{2}-207 \right) {t}^{5}+ \left( -1792\,{y}^{6}-1554\,{y}^{4}+936\,{y}^{2}+927 \right) { t}^{4}-2\,y \left( 360\,{y}^{6}+320\,{y}^{4}-557\,{y}^{2}-429 \right) { t}^{3}+ \left( -16\,{y}^{6}+189\,{y}^{4}+230\,{y}^{2}-9 \right) {t}^{2} +2\,y \left( {y}^{2}-3 \right) t-{y}^{2} \left( {y}^{2}+2 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( 2\,y+{ y}^{-1} \right) \left( 2\,y+3\,{y}^{-1} \right) {t}^{2} \right) ^{-5/2 }}\,{\rm d}t\,{\rm d}t\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}+{y}^{2}+x+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1-4\, \left( x+{x}^{-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2} \right) ^{3/2}} \left( 2+24\,\int \!{\frac {\sqrt {1-4\, \left( x+{x}^ {-1} \right) \left( x+1+{x}^{-1} \right) {t}^{2}}t}{ \left( 16\,{t}^{2 }+1 \right) ^{7/2}{x}^{2}} \left( 10\, \left( 2\,{x}^{2}+x+2 \right) {t }^{2} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( {x}^{2}+x+1 \right) \left( 16\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+1}{ t \left( t-y \right) }\int \!\!\!\int \!-2\,y+\int \!{(4\,t{y}^{4}-3\,t -y) \left( -6\,{y}^{-1}+\int \!-24\,{\frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{11/2}{y}^{2} \left( 4\,t{y}^{4}-3\,t-y \right) ^{2}} \left( 1-2\,{\frac {t}{y}}- \left( 2\,y+{y}^{-1} \right) \left( 2\,y+3\,{y}^{ -1} \right) {t}^{2} \right) ^{3/2} \left( \left( 16\,{t}^{2}+1 \right) \left( 10\, \left( 10\,{y}^{2}+9 \right) \left( 16\,{y}^{4}+ 14\,{y}^{2}-3 \right) {t}^{6}-10\,y \left( 4\,{y}^{2}+3 \right) \left( 8\,{y}^{4}+12\,{y}^{2}-3 \right) {t}^{5}+ \left( -576\,{y}^{6}- 944\,{y}^{4}+69\,{y}^{2}+351 \right) {t}^{4}-4\,y \left( 40\,{y}^{6}+20 \,{y}^{4}-83\,{y}^{2}-66 \right) {t}^{3}+ \left( -16\,{y}^{6}+90\,{y}^{ 4}+102\,{y}^{2}-9 \right) {t}^{2}+2\,y \left( {y}^{2}-3 \right) t-{y}^{ 2} \left( {y}^{2}+2 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 464\,{y}^{6}+496\,{y}^{4}-276\,{y}^{2}-324 \right) {t}^{6}+8\,y \left( 190\,{y}^{6}+270\,{y}^{4}-141\,{y}^{2}-207 \right) {t}^{5}+ \left( -1792\,{y}^{6}-1554\,{y}^{4}+936\,{y}^{2}+927 \right) { t}^{4}-2\,y \left( 360\,{y}^{6}+320\,{y}^{4}-557\,{y}^{2}-429 \right) { t}^{3}+ \left( -16\,{y}^{6}+189\,{y}^{4}+230\,{y}^{2}-9 \right) {t}^{2} +2\,y \left( {y}^{2}-3 \right) t-{y}^{2} \left( {y}^{2}+2 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( 2\,y+{ y}^{-1} \right) \left( 2\,y+3\,{y}^{-1} \right) {t}^{2} \right) ^{-5/2 }}\,{\rm d}t\,{\rm d}t\,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{ \frac {1}{ \left( 16\,{t}^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+1 \right) {t}^{2}}{ \left( 16\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) } 14 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( 12\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( t-1 \right) {t}^{2}}\int \!\!\!\int \!-2+\int \!6\,{ \frac {1-t}{ \left( -35\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 1-\int \!4\,{\frac { \left( -35\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 12\,{t }^{2}+1 \right) ^{11/2} \left( t-1 \right) ^{2}} \left( \left( 12\,{t} ^{2}+1 \right) \left( -10\,{t}^{6}+289\,{t}^{5}-515\,{t}^{4}-1802\,{t} ^{3}-487\,{t}^{2}+8\,t+6 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} +6\, \left( 590\,{t}^{6}-571\,{t}^{5}-103\,{t}^{4}+3079\,{t}^{3}+769\,{ t}^{2}-14\,t-6 \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -20\,{t}^{2} -4\,t+1 \right) \left( 12\,{t}^{2}+1 \right) ^{3/2}} \left( \left( 12 \,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -t \left( 8\,t+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( t-1 \right) t}\int \!-1+\int \!{\frac { \left( 2\,t+ 1 \right) \left( 5\,t+1 \right) }{ \left( -35\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( -2+\int \!6\,{\frac { \left( -35\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 5\,t+1 \right) \left( 12\,{t}^{2}+1 \right) ^ {9/2} \left( 2\,t+1 \right) ^{2}} \left( \left( 12\,{t}^{2}+1 \right) \left( 1808\,{t}^{5}+1084\,{t}^{4}+540\,{t}^{3}+107\,{t}^{2}-132\,t-5 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -14\,t \left( 636\,{t}^{6}+2104\,{t}^{5}+811\,{t}^{4}-500\,{t}^{3}-403 \,{t}^{2}-55\,t-10 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{1 \left( 2+\int \!6\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{x}^{2}}\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) } \left( 30\,{t}^{3} \left( t{x}^{2}+2\,{x}^{2}+t+x+2 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( \left( 3\,{x}^{2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) \right) ^{-3/2} }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{ \left( t-y \right) {t}^{2}}\int \!\!\!\int \!-2\,y+\int \!{ ( \left( 4\,{y}^{4}+4\,{y}^{3}-4\,y-3 \right) t-y) \left( -6\,{y}^{-1}+ \int \!-24\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{11/2}{y}^{2} \left( \left( 4\,{y}^{4}+4\,{y}^{3}-4\,y-3 \right) t-y \right) ^{2}} \left( 1-2\,{\frac {t}{y}}- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2 }-{y}^{-2} \right) {t}^{2} \right) ^{3/2} \left( - \left( 12\,{t}^{2}+1 \right) \left( \left( 448\,{y}^{7}-128\,{y}^{6}+704\,{y}^{5}-384\,{y }^{4}-336\,{y}^{3}-764\,{y}^{2}+108\,y+342 \right) {t}^{6}+ \left( 864 \,{y}^{7}+1776\,{y}^{6}+1432\,{y}^{5}+412\,{y}^{4}-1380\,{y}^{3}-1588\, {y}^{2}-1038\,y-189 \right) {t}^{5}+ \left( 528\,{y}^{7}+3072\,{y}^{6}+ 3312\,{y}^{5}+834\,{y}^{4}-2874\,{y}^{3}-3371\,{y}^{2}-1638\,y-378 \right) {t}^{4}+2\,y \left( 120\,{y}^{6}+480\,{y}^{5}+377\,{y}^{4}-234 \,{y}^{3}-864\,{y}^{2}-624\,y-156 \right) {t}^{3}+ \left( 16\,{y}^{7}+ 64\,{y}^{6}-14\,{y}^{5}-192\,{y}^{4}-294\,{y}^{3}-112\,{y}^{2}+36\,y+9 \right) {t}^{2}-y \left( 2\,{y}^{3}+9\,{y}^{2}-13\,y-6 \right) t+{y}^{ 2} \left( y+2 \right) \left( {y}^{2}+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 448\,{y}^{7}+832\,{y}^{6}+1024\,{y}^{5}+256\,{y}^{4 }-616\,{y}^{3}-844\,{y}^{2}-492\,y-18 \right) {t}^{6}+ \left( -352\,{y} ^{7}-2368\,{y}^{6}-3216\,{y}^{5}-2100\,{y}^{4}+1312\,{y}^{3}+2940\,{y}^ {2}+2286\,y+927 \right) {t}^{5}+ \left( -2048\,{y}^{7}-5792\,{y}^{6}- 6956\,{y}^{5}-2206\,{y}^{4}+4992\,{y}^{3}+6849\,{y}^{2}+4140\,y+918 \right) {t}^{4}+ \left( -816\,{y}^{7}-1584\,{y}^{6}-1260\,{y}^{5}+900 \,{y}^{4}+2677\,{y}^{3}+2325\,{y}^{2}+846\,y-9 \right) {t}^{3}+ \left( -16\,{y}^{7}-64\,{y}^{6}+39\,{y}^{5}+314\,{y}^{4}+328\,{y}^{3}+219\,{y} ^{2}-42\,y-9 \right) {t}^{2}-y \left( {y}^{4}-8\,{y}^{2}+15\,y+6 \right) t-{y}^{2} \left( y+2 \right) \left( {y}^{2}+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{-5 /2}}\,{\rm d}t\,{\rm d}t\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}{y}^{2}+{x}^{2}y+{x}^{2}+{y}^{2}+x+y+1 \right) } \left( xy-{\frac {{x}^{2}+x+1}{t}\int \!\!\!\int \!{1 \left( 2+\int \!6\,{\frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{7/2}{x}^ {2}}\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x }^{-1} \right) t \right) } \left( 30\,{t}^{3} \left( t{x}^{2}+2\,{x}^{2 }+t+x+2 \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} - \left( 12\,{t}^{2}+1 \right) \left( \left( 3\,{x}^{2}+4\,x+3 \right) t+x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1- \left( x+{x}^{-1} \right) t \left( 2+ \left( 3\,x+4+3\,{x}^{-1} \right) t \right) \right) ^{-3/2} }\,{\rm d}t\,{\rm d}t}-{\frac {{y}^{2}+y+1}{t \left( t-y \right) }\int \!\!\!\int \!-2\,y+\int \!{( \left( 4\,{y}^{4}+4\,{y}^{3}-4\,y-3 \right) t-y) \left( -6\,{y}^{-1}+\int \!-24\,{\frac {1}{ \left( 12\,{t }^{2}+1 \right) ^{11/2}{y}^{2} \left( \left( 4\,{y}^{4}+4\,{y}^{3}-4\, y-3 \right) t-y \right) ^{2}} \left( 1-2\,{\frac {t}{y}}- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{3/ 2} \left( - \left( 12\,{t}^{2}+1 \right) \left( \left( 448\,{y}^{7}- 128\,{y}^{6}+704\,{y}^{5}-384\,{y}^{4}-336\,{y}^{3}-764\,{y}^{2}+108\,y +342 \right) {t}^{6}+ \left( 864\,{y}^{7}+1776\,{y}^{6}+1432\,{y}^{5}+ 412\,{y}^{4}-1380\,{y}^{3}-1588\,{y}^{2}-1038\,y-189 \right) {t}^{5}+ \left( 528\,{y}^{7}+3072\,{y}^{6}+3312\,{y}^{5}+834\,{y}^{4}-2874\,{y} ^{3}-3371\,{y}^{2}-1638\,y-378 \right) {t}^{4}+2\,y \left( 120\,{y}^{6} +480\,{y}^{5}+377\,{y}^{4}-234\,{y}^{3}-864\,{y}^{2}-624\,y-156 \right) {t}^{3}+ \left( 16\,{y}^{7}+64\,{y}^{6}-14\,{y}^{5}-192\,{y}^{ 4}-294\,{y}^{3}-112\,{y}^{2}+36\,y+9 \right) {t}^{2}-y \left( 2\,{y}^{3 }+9\,{y}^{2}-13\,y-6 \right) t+{y}^{2} \left( y+2 \right) \left( {y}^{ 2}+1 \right) \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} -6\, \left( \left( 448\,{y}^{7}+832\,{y}^{6}+1024\,{y}^{5}+256\,{y}^{4 }-616\,{y}^{3}-844\,{y}^{2}-492\,y-18 \right) {t}^{6}+ \left( -352\,{y} ^{7}-2368\,{y}^{6}-3216\,{y}^{5}-2100\,{y}^{4}+1312\,{y}^{3}+2940\,{y}^ {2}+2286\,y+927 \right) {t}^{5}+ \left( -2048\,{y}^{7}-5792\,{y}^{6}- 6956\,{y}^{5}-2206\,{y}^{4}+4992\,{y}^{3}+6849\,{y}^{2}+4140\,y+918 \right) {t}^{4}+ \left( -816\,{y}^{7}-1584\,{y}^{6}-1260\,{y}^{5}+900 \,{y}^{4}+2677\,{y}^{3}+2325\,{y}^{2}+846\,y-9 \right) {t}^{3}+ \left( -16\,{y}^{7}-64\,{y}^{6}+39\,{y}^{5}+314\,{y}^{4}+328\,{y}^{3}+219\,{y} ^{2}-42\,y-9 \right) {t}^{2}-y \left( {y}^{4}-8\,{y}^{2}+15\,y+6 \right) t-{y}^{2} \left( y+2 \right) \left( {y}^{2}+1 \right) \right) {t}^{2} {{}_{2}F_{1}\!\left(11/4,{\frac{13}{4}};4;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) \left( 1-2\,{\frac {t}{y}}- \left( 4\, \left( y+1+{y}^{-1} \right) ^{2}-{y}^{-2} \right) {t}^{2} \right) ^{-5 /2}}\,{\rm d}t\,{\rm d}t\,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{ \frac {1}{ \left( 12\,{t}^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( {t}^{2}+t+1 \right) {t}^{2}}{ \left( 12\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) }$
15 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {\int \!2\,t{{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)} \,{\rm d}t}{{t}^{2}}} 01\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle {\frac {\sqrt { \left( 1-3\,t \right) \left( t+1 \right) } \left( 2\,t -1 \right) }{{t}^{3}}\int \!{\frac {t}{ \left( \left( 1-3\,t \right) \left( t+1 \right) \right) ^{3/2}} \left( 1+{\frac { \left( 2\,t-1 \right) {{}_{2}F_{1}\!\left(-1/4,1/4;1;64\,{t}^{4}\right)}+t \left( 8 \,{t}^{2}+t-1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)} }{ \left( 2\,t-1 \right) ^{2}}} \right) }\,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {-8\,{t}^{2}+1}} \left( 2- \int \!8\,{\frac {t{{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{ \sqrt {-8\,{t}^{2}+1}}}\,{\rm d}t \right) }\,{\rm d}t} 11\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle 1/4\,{\frac {1-2\,t}{{t}^{2}} \left( 1-{\frac {\sqrt { \left( 1-3\,t \right) \left( t+1 \right) }}{1-3\,t} \left( 1-\int \!2\,{\frac { \left( -8\,{t}^{3}-6\,{t}^{2}+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}+4\,{t}^{3} \left( 4\, {t}^{2}-7\,t+1 \right) {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{ \left( 1-2\,t \right) ^{2}\sqrt { \left( 1-3\,t \right) \left( t+1 \right) } \left( t+1 \right) }}\,{\rm d}t \right) } \right) }$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!{\frac {t}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}} \left( 2-\int \!8\,{\frac {t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}x}}\,{\rm d}t \right) }\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)}$$\displaystyle {\frac {1}{{t}^{3}}\sqrt {1-2\,ty+ \left( {y}^{2}-4\,{y}^{-2} \right) { t}^{2}} \left( 2\,t-{y}^{-1} \right) \int \!{t \left( 1+{\frac {y \left( 2\,ty-1 \right) {{}_{2}F_{1}\!\left(-1/4,1/4;1;64\,{t}^{4}\right)}+t \left( t{y}^{3}+8 \,{t}^{2}-{y}^{2} \right) {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)}}{ \left( 2\,ty-1 \right) ^{2}y}} \right) \left( 1-2\,ty+ \left( {y}^{2}-4\,{y}^{-2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{t}^{4}\right)} $$\displaystyle {\frac {1}{xy-t \left( x{y}^{2}+{x}^{2}+1 \right) } \left( xy-{\frac {{ x}^{2}+1}{t}\int \!{\frac {t}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t }^{2}}} \left( 2-\int \!8\,{\frac {t {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{t}^{4}\right)}}{\sqrt {1-4\, \left( x+{x}^{-1} \right) {t}^{2}}x}}\,{\rm d}t \right) }\,{\rm d}t}-{\frac {1 }{{t}^{2}}\sqrt {1-2\,ty+ \left( {y}^{2}-4\,{y}^{-2} \right) {t}^{2}} \left( 2\,t-{y}^{-1} \right) \int \!{t \left( 1+{\frac {y \left( 2\,ty -1 \right) {{}_{2}F_{1}\!\left(-1/4,1/4;1;64\,{t}^{4}\right)}+t \left( t{y}^{3}+8\,{t}^{2}-{y}^{2} \right) {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)}}{ \left( 2\,ty-1 \right) ^{2}y}} \right) \left( 1-2\,ty+ \left( {y}^{2}-4\,{y}^{-2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t}+{\frac {\int \!2\,t {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{t}^{4}\right)}\,{\rm d}t}{t}} \right) } 16 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!2\,{\frac {1}{ \left( -4\,{t}^{2}+ 1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t}$
01$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{3}}\int \!\!\!\int \!\!\!\int \!{\frac {1-9\,t}{ \left( -15\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( 6+12\,\int \!{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( -4\,{t}^{2}+1 \right) ^{7/2} \left( t+1 \right) ^{2} \left( 1-9\,t \right) ^{2}} \left( \left( -816\,{t}^{6}-832\,{t}^{5}+288\,{t}^{4}+128\,{t}^{3}+71 \,{t}^{2}-18\,t+5 \right) {{}_{2}F_{1}\!\left(5/4,7/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} + \left( -744\,{t}^{6}-336\,{t}^{5}+110\,{t}^{4}+14\,{t}^{3}+51\,{t}^{2 }+8\,t-1 \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t\,{\rm d}t} 10\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!-1/2+{\frac {1}{ \left( -4\,{t}^{2}+1 \right) ^{3/2}} \left( \left( 1/2+2\,t+6\,{t}^{2} \right) {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -6\,{t}^{2} \left( t+1 \right) {{}_{2}F_{1}\!\left(1/4,3/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t}$
11$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {t \left( 3\,t+1 \right) }{ \left( -15\,{t}^{2}-2\,t+1 \right) ^{5/2}} \left( -4+\int \!2\,{\frac { \left( -15\,{t}^{2}-2\,t+1 \right) ^{3/2}}{ \left( 3\,t+1 \right) \left( -4\,{t}^{2}+1 \right) ^{9/2}{t}^{2}} \left( \left( 4\,{t}^{2}- 1 \right) \left( 92\,{t}^{4}+76\,{t}^{3}+43\,{t}^{2}+6\,t+1 \right) {{}_{2}F_{1}\!\left(7/4,9/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +14\,{t}^{3} \left( 10\,t+1 \right) \left( 18\,{t}^{3}+7\,{t}^{2}+3\,t -1 \right) {{}_{2}F_{1}\!\left(9/4,11/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t} x0\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{{t}^{2}}\int \!\!\!\int \!{\frac {1}{ \left( 1- \left( x+{x} ^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) }}{ \left( - 4\,{t}^{2}+1 \right) ^{7/2}{x}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( \left( {x}^{2}-4\,x+1 \right) t-x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -10\,{t}^{3} \left( t{x}^{2}+t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} $$\displaystyle {\frac {ty-t-1}{{t}^{2}}\int \!{\frac { \left( 5\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{{t}^{2} \left( ty-t-1 \right) ^{2}}\int \!{\frac { \left( ty-t-1 \right) \left( \left( 3\, {y}^{5}-6\,{y}^{4}-18\,{y}^{3}-24\,{y}^{2}-16\,y-8 \right) {t}^{3}-3\,y \left( 3\,{y}^{3}-{y}^{2}-2\,y-2 \right) {t}^{2}+6\,t{y}^{3}-{y}^{2} \right) t}{ \left( \left( 5\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y \right) ^{2}}\int \!6\,{\frac { \left( 5\,{y}^{ 2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{ \left( -4\,{t}^ {2}+1 \right) ^{7/2} \left( \left( 3\,{y}^{5}-6\,{y}^{4}-18\,{y}^{3}- 24\,{y}^{2}-16\,y-8 \right) {t}^{3}-3\,y \left( 3\,{y}^{3}-{y}^{2}-2\,y -2 \right) {t}^{2}+6\,t{y}^{3}-{y}^{2} \right) ^{2}{y}^{2}}\sqrt {1-2\, ty- \left( \left( 2+2\,{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( 4\,y \left( 3\,{y}^{6}-18 \,{y}^{5}+2\,{y}^{4}+44\,{y}^{3}-36\,{y}^{2}-32\,y+40 \right) {t}^{6}+ \left( -60\,{y}^{6}+120\,{y}^{5}+88\,{y}^{4}-512\,{y}^{3}-464\,{y}^{2} +128\,y+32 \right) {t}^{5}+8\,y \left( 5\,{y}^{4}-10\,{y}^{3}-23\,{y}^{ 2}-22\,y-4 \right) {t}^{4}-2\,{y}^{2} \left( y+1 \right) \left( 5\,{y} ^{2}+2\,y-2 \right) {t}^{3}+2\,{y}^{2} \left( 5\,y+8 \right) \left( y+ 1 \right) {t}^{2}+t{y}^{5}-{y}^{4} \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +10\, \left( \left( 32\,{y}^{5}+32\,{y}^{4}-32\,{y}^{3}-16\,{y}^{2}-32 \,y-32 \right) {t}^{5}+y \left( 3\,{y}^{6}-6\,{y}^{5}+14\,{y}^{4}+36\,{ y}^{3}+4\,{y}^{2}+80\,y+56 \right) {t}^{4}-{y}^{2} \left( 9\,{y}^{4}-18 \,{y}^{3}-40\,{y}^{2}+32\,y+28 \right) {t}^{3}+2\,{y}^{2} \left( y+1 \right) \left( 4\,{y}^{2}-7\,y-8 \right) {t}^{2}-t{y}^{5}+{y}^{4} \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \left( 1-2\,ty- \left( \left( 2+2\,{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t} \,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/4,3/4;1;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}y+x{y}^{2}+{x}^{2}+y+1 \right) } \left( x y-{\frac {{x}^{2}+1}{t}\int \!\!\!\int \!{\frac {1}{ \left( 1- \left( x +{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) \right) ^{3/2}} \left( 2+\int \!6\,{\frac {\sqrt {1- \left( x+{x}^{-1} \right) t \left( 2- \left( x-4+{x}^{-1} \right) t \right) }}{ \left( - 4\,{t}^{2}+1 \right) ^{7/2}{x}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( \left( {x}^{2}-4\,x+1 \right) t-x \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} -10\,{t}^{3} \left( t{x}^{2}+t+x \right) {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \right) }\,{\rm d}t\,{\rm d}t}-{\frac { \left( y+1 \right) \left( ty-t-1 \right) }{t}\int \!{\frac { \left( 5\,{y}^{2}+4 \,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{{t}^{2} \left( ty-t- 1 \right) ^{2}}\int \!{\frac { \left( ty-t-1 \right) \left( \left( 3 \,{y}^{5}-6\,{y}^{4}-18\,{y}^{3}-24\,{y}^{2}-16\,y-8 \right) {t}^{3}-3 \,y \left( 3\,{y}^{3}-{y}^{2}-2\,y-2 \right) {t}^{2}+6\,t{y}^{3}-{y}^{2 } \right) t}{ \left( \left( 5\,{y}^{2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y \right) ^{2}}\int \!6\,{\frac { \left( 5\,{y}^{ 2}+4\,y+2 \right) {t}^{2}+2\,y \left( y-1 \right) t-y}{ \left( -4\,{t}^ {2}+1 \right) ^{7/2} \left( \left( 3\,{y}^{5}-6\,{y}^{4}-18\,{y}^{3}- 24\,{y}^{2}-16\,y-8 \right) {t}^{3}-3\,y \left( 3\,{y}^{3}-{y}^{2}-2\,y -2 \right) {t}^{2}+6\,t{y}^{3}-{y}^{2} \right) ^{2}{y}^{2}}\sqrt {1-2\, ty- \left( \left( 2+2\,{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2}} \left( \left( -4\,{t}^{2}+1 \right) \left( 4\,y \left( 3\,{y}^{6}-18 \,{y}^{5}+2\,{y}^{4}+44\,{y}^{3}-36\,{y}^{2}-32\,y+40 \right) {t}^{6}+ \left( -60\,{y}^{6}+120\,{y}^{5}+88\,{y}^{4}-512\,{y}^{3}-464\,{y}^{2} +128\,y+32 \right) {t}^{5}+8\,y \left( 5\,{y}^{4}-10\,{y}^{3}-23\,{y}^{ 2}-22\,y-4 \right) {t}^{4}-2\,{y}^{2} \left( y+1 \right) \left( 5\,{y} ^{2}+2\,y-2 \right) {t}^{3}+2\,{y}^{2} \left( 5\,y+8 \right) \left( y+ 1 \right) {t}^{2}+t{y}^{5}-{y}^{4} \right) {{}_{2}F_{1}\!\left(5/4,7/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} +10\, \left( \left( 32\,{y}^{5}+32\,{y}^{4}-32\,{y}^{3}-16\,{y}^{2}-32 \,y-32 \right) {t}^{5}+y \left( 3\,{y}^{6}-6\,{y}^{5}+14\,{y}^{4}+36\,{ y}^{3}+4\,{y}^{2}+80\,y+56 \right) {t}^{4}-{y}^{2} \left( 9\,{y}^{4}-18 \,{y}^{3}-40\,{y}^{2}+32\,y+28 \right) {t}^{3}+2\,{y}^{2} \left( y+1 \right) \left( 4\,{y}^{2}-7\,y-8 \right) {t}^{2}-t{y}^{5}+{y}^{4} \right) {t}^{3} {{}_{2}F_{1}\!\left(7/4,9/4;3;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} \right) }\,{\rm d}t \left( 1-2\,ty- \left( \left( 2+2\,{y}^{-1} \right) ^{2}-{y}^{2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t} \,{\rm d}t}+{\frac {1}{t}\int \!\!\!\int \!2\,{\frac {1}{ \left( -4\,{t }^{2}+1 \right) ^{3/2}} {{}_{2}F_{1}\!\left(3/4,5/4;2;64\,{\frac { \left( t+1 \right) {t}^{3}}{ \left( -4\,{t}^{2}+1 \right) ^{2}}}\right)} }\,{\rm d}t\,{\rm d}t} \right) }$
17 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)} $$\displaystyle 1/3\,{\frac {{{}_{2}F_{1}\!\left(-2/3,-1/3;2;27\,{t}^{3}\right)}-1}{{t} ^{3}}} 01\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)}$$\displaystyle {\frac {\sqrt {t+1} \left( 1-3\,t \right) ^{3/2}}{{t}^{3}}\int \!1/3\,{ \frac {{{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}-1+4\,t \left( {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}-1 \right) +9 \,{t}^{2}}{ \left( t+1 \right) ^{3/2} \left( 1-3\,t \right) ^{5/2}}} \,{\rm d}t}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)} $$\displaystyle {\frac {\sqrt {-3\,{t}^{2}-2\,t+1} \left( 3\,t-1 \right) }{{t}^{3}} \int \!1/3\,{\frac {27\,{t}^{3}+3\,{t}^{2}-t-1- \left( 3\,t-1 \right) {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}-4\,t \left( 3\,t-1 \right) {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}}{ \left( -3 \,{t}^{2}-2\,t+1 \right) ^{3/2} \left( 3\,t-1 \right) ^{2}}}\,{\rm d}t} 11\displaystyle {\it none}$$\displaystyle 1/2\,{\frac {1-t-\sqrt {-3\,{t}^{2}-2\,t+1}}{{t}^{2}}}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)} $$\displaystyle {\frac {t{x}^{3}+2\,t-x}{{t}^{3}}\sqrt {1-2\,{\frac {t}{x}}- \left( 4\, x-{x}^{-2} \right) {t}^{2}}\int \!1/3\,{\frac {6\,{x}^{5}{t}^{3}+21\,{x }^{2}{t}^{3}-{x}^{3}{t}^{2}+4\,{t}^{2}-tx-{x}^{2}- \left( 3\,t-x \right) x{{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}-4\,t \left( 2\,t{x}^{3}+t-x \right) {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}}{{x}^{2} \left( t{x} ^{3}+2\,t-x \right) ^{2}} \left( 1-2\,{\frac {t}{x}}- \left( 4\,x-{x}^{ -2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)}$$\displaystyle {\frac {2\,t{y}^{3}-{y}^{2}+t}{{t}^{3}}\sqrt {1-2\,ty+ \left( {y}^{2}-4 \,{y}^{-1} \right) {t}^{2}}\int \!1/3\,{\frac {15\,{t}^{3}{y}^{4}+12\,{ t}^{3}y-13\,{t}^{2}{y}^{3}-8\,{t}^{2}+t{y}^{2}+y+ \left( 3\,ty-1 \right) y{{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}+4\, \left( t{y}^{3}-{y}^{2}+2\,t \right) t {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}}{y \left( 2\,t{y}^{3 }-{y}^{2}+t \right) ^{2}} \left( 1-2\,ty+ \left( {y}^{2}-4\,{y}^{-1} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\,{t}^{3}\right)} $$\displaystyle {\frac {1}{xy-t \left( x{y}^{2}+{x}^{2}+y \right) } \left( xy-{\frac {{ x}^{2} \left( t{x}^{3}+2\,t-x \right) }{{t}^{2}}\sqrt {1-2\,{\frac {t}{ x}}- \left( 4\,x-{x}^{-2} \right) {t}^{2}}\int \!1/3\,{\frac {6\,{x}^{5 }{t}^{3}+21\,{x}^{2}{t}^{3}-{x}^{3}{t}^{2}+4\,{t}^{2}-tx-{x}^{2}- \left( 3\,t-x \right) x {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}-4\,t \left( 2\,t{x} ^{3}+t-x \right) {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}}{{x }^{2} \left( t{x}^{3}+2\,t-x \right) ^{2}} \left( 1-2\,{\frac {t}{x}}- \left( 4\,x-{x}^{-2} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t}-{ \frac {y \left( 2\,t{y}^{3}-{y}^{2}+t \right) }{{t}^{2}}\sqrt {1-2\,ty+ \left( {y}^{2}-4\,{y}^{-1} \right) {t}^{2}}\int \!1/3\,{\frac {15\,{t} ^{3}{y}^{4}+12\,{t}^{3}y-13\,{t}^{2}{y}^{3}-8\,{t}^{2}+t{y}^{2}+y+ \left( 3\,ty-1 \right) y {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\,{t}^{3}\right)}+4\, \left( t{y}^{3} -{y}^{2}+2\,t \right) t {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\,{t}^{3}\right)}}{y \left( 2\,t{y}^{3 }-{y}^{2}+t \right) ^{2}} \left( 1-2\,ty+ \left( {y}^{2}-4\,{y}^{-1} \right) {t}^{2} \right) ^{-3/2}}\,{\rm d}t} \right) } 18 Your browser does not support the HTML5 canvas tag. 00\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)}$$\displaystyle 1/3\,{\frac {1}{{t}^{3}} \left( {\frac { \left( 2\,t+1 \right) \left( {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} +4\,t {{}_{2}F_{1}\!\left(-1/3,1/3;2;27\, \left( 2\,t+1 \right) {t}^{2}\right)} \right) }{3\,t+1}}-1-3\,t-5\,{t}^{2} \right) }$
01$\displaystyle {\it none} $$\displaystyle 1/32\,{\frac { \left( 1-6\,t \right) \sqrt {-12\,{t}^{2}-4\,t+1}-4\,{t} ^{2}+8\,t-1}{{t}^{3}}} 10\displaystyle {\it none}$$\displaystyle 1/32\,{\frac { \left( 1-6\,t \right) \sqrt {-12\,{t}^{2}-4\,t+1}-4\,{t} ^{2}+8\,t-1}{{t}^{3}}}$
11$\displaystyle {\it none} $$\displaystyle 1/8\,{\frac {1-2\,t-\sqrt {-12\,{t}^{2}-4\,t+1}}{{t}^{2}}} x0\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)}$$\displaystyle {\frac {\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x }^{-1} \right) ^{2}-12 \right) {t}^{2}} \left( \left( {x}^{4}+2\,{x}^{ 3}+2\,x+1 \right) t-{x}^{2} \right) }{{t}^{3}}\int \!1/6\,{\frac {3\, \left( 1+x \right) \left( 4\,{x}^{5}+4\,{x}^{4}-5\,{x}^{3}+14\,{x}^{2 }-x+2 \right) {t}^{3}- \left( x+2 \right) \left( 4\,{x}^{4}+2\,{x}^{3} -3\,{x}^{2}+5\,x-2 \right) {t}^{2}+x \left( x-1 \right) \left( 3\,{x}^ {2}+2\,x+3 \right) t-{x}^{2} \left( x-1 \right) +1/3\, \left( x-1 \right) \left( \left( 2\,t{x}^{4}+4\,t{x}^{3}+12\,t{x}^{2}-2\,{x}^{3 }+4\,tx+2\,t-2\,x \right) {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} + \left( 6\,t-1 \right) \left( 2\,t{x}^{4}+t{x}^{3}-2\,{x}^{3}+tx-3\,{ x}^{2}+2\,t-2\,x \right) {{}_{2}F_{1}\!\left(-1/3,1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} \right) }{ \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x- 2+{x}^{-1} \right) ^{2}-12 \right) {t}^{2} \right) ^{3/2} \left( \left( {x}^{4}+2\,{x}^{3}+2\,x+1 \right) t-{x}^{2} \right) ^{2}{x}^{2} }}\,{\rm d}t}$
0y$\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} $$\displaystyle {\frac {\sqrt {1-2\, \left( y+{y}^{-1} \right) t+ \left( \left( y-2+{y }^{-1} \right) ^{2}-12 \right) {t}^{2}} \left( \left( {y}^{4}+2\,{y}^{ 3}+2\,y+1 \right) t-{y}^{2} \right) }{{t}^{3}}\int \!1/6\,{\frac {3\, \left( y+1 \right) \left( 4\,{y}^{5}+4\,{y}^{4}-5\,{y}^{3}+14\,{y}^{2 }-y+2 \right) {t}^{3}- \left( y+2 \right) \left( 4\,{y}^{4}+2\,{y}^{3} -3\,{y}^{2}+5\,y-2 \right) {t}^{2}+y \left( y-1 \right) \left( 3\,{y}^ {2}+2\,y+3 \right) t-{y}^{2} \left( y-1 \right) +1/3\, \left( y-1 \right) \left( \left( 2\,t{y}^{4}+4\,t{y}^{3}+12\,t{y}^{2}-2\,{y}^{3 }+4\,ty+2\,t-2\,y \right) {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} + \left( 6\,t-1 \right) \left( 2\,t{y}^{4}+t{y}^{3}-2\,{y}^{3}+ty-3\,{ y}^{2}+2\,t-2\,y \right) {{}_{2}F_{1}\!\left(-1/3,1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} \right) }{ \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( \left( y- 2+{y}^{-1} \right) ^{2}-12 \right) {t}^{2} \right) ^{3/2} \left( \left( {y}^{4}+2\,{y}^{3}+2\,y+1 \right) t-{y}^{2} \right) ^{2}{y}^{2} }}\,{\rm d}t} xy\displaystyle {{}_{2}F_{1}\!\left(1/3,2/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}y+x{y}^{2}+{x}^{2}+{y}^{2}+x+y \right) } \left( xy-{\frac { \left( 1+x \right) x\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x-2+{x}^{-1} \right) ^{2}-12 \right) {t}^{2} } \left( \left( {x}^{4}+2\,{x}^{3}+2\,x+1 \right) t-{x}^{2} \right) }{ {t}^{2}}\int \!1/6\,{\frac {3\, \left( 1+x \right) \left( 4\,{x}^{5}+4 \,{x}^{4}-5\,{x}^{3}+14\,{x}^{2}-x+2 \right) {t}^{3}- \left( x+2 \right) \left( 4\,{x}^{4}+2\,{x}^{3}-3\,{x}^{2}+5\,x-2 \right) {t}^{2 }+x \left( x-1 \right) \left( 3\,{x}^{2}+2\,x+3 \right) t-{x}^{2} \left( x-1 \right) +1/3\, \left( x-1 \right) \left( \left( 2\,t{x}^{ 4}+4\,t{x}^{3}+12\,t{x}^{2}-2\,{x}^{3}+4\,tx+2\,t-2\,x \right) {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} + \left( 6\,t-1 \right) \left( 2\,t{x}^{4}+t{x}^{3}-2\,{x}^{3}+tx-3\,{ x}^{2}+2\,t-2\,x \right) {{}_{2}F_{1}\!\left(-1/3,1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} \right) }{ \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( \left( x- 2+{x}^{-1} \right) ^{2}-12 \right) {t}^{2} \right) ^{3/2} \left( \left( {x}^{4}+2\,{x}^{3}+2\,x+1 \right) t-{x}^{2} \right) ^{2}{x}^{2} }}\,{\rm d}t}-{\frac { \left( y+1 \right) y\sqrt {1-2\, \left( y+{y}^{- 1} \right) t+ \left( \left( y-2+{y}^{-1} \right) ^{2}-12 \right) {t}^{ 2}} \left( \left( {y}^{4}+2\,{y}^{3}+2\,y+1 \right) t-{y}^{2} \right) }{{t}^{2}}\int \!1/6\,{\frac {3\, \left( y+1 \right) \left( 4\,{y}^{5} +4\,{y}^{4}-5\,{y}^{3}+14\,{y}^{2}-y+2 \right) {t}^{3}- \left( y+2 \right) \left( 4\,{y}^{4}+2\,{y}^{3}-3\,{y}^{2}+5\,y-2 \right) {t}^{2 }+y \left( y-1 \right) \left( 3\,{y}^{2}+2\,y+3 \right) t-{y}^{2} \left( y-1 \right) +1/3\, \left( y-1 \right) \left( \left( 2\,t{y}^{ 4}+4\,t{y}^{3}+12\,t{y}^{2}-2\,{y}^{3}+4\,ty+2\,t-2\,y \right) {{}_{2}F_{1}\!\left(-2/3,-1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} + \left( 6\,t-1 \right) \left( 2\,t{y}^{4}+t{y}^{3}-2\,{y}^{3}+ty-3\,{ y}^{2}+2\,t-2\,y \right) {{}_{2}F_{1}\!\left(-1/3,1/3;1;27\, \left( 2\,t+1 \right) {t}^{2}\right)} \right) }{ \left( 1-2\, \left( y+{y}^{-1} \right) t+ \left( \left( y- 2+{y}^{-1} \right) ^{2}-12 \right) {t}^{2} \right) ^{3/2} \left( \left( {y}^{4}+2\,{y}^{3}+2\,y+1 \right) t-{y}^{2} \right) ^{2}{y}^{2} }}\,{\rm d}t} \right) }$
19 Your browser does not support the HTML5 canvas tag. 00$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle 1/4\,{\frac {6\,{t}^{2}+1- {{}_{2}F_{1}\!\left(-3/2,-1/2;2;16\,{t}^{2}\right)}}{{t}^{4}}} 01\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/2\,{\frac {1-{{}_{2}F_{1}\!\left(-1/2,1/2;2;16\,{t}^{2}\right)}}{{t}^ {2}}}$
10$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle 1/4\,{\frac {{{}_{2}F_{1}\!\left(-1/2,-1/2;1;16\,{t}^{2}\right)}-2\,t {{}_{2}F_{1}\!\left(-1/2,1/2;2;16\,{t}^{2}\right)}-4\,{t}^{2}+2\,t-1}{{ t}^{3}}} 11\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/4\,{\frac {1-2\,t-{{}_{2}F_{1}\!\left(-1/2,1/2;1;16\,{t}^{2}\right)}+ 2\,t{{}_{2}F_{1}\!\left(1/2,1/2;2;16\,{t}^{2}\right)}}{{t}^{2}}}$
x0$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)} $$\displaystyle 1/2\,{\frac {2\,t{x}^{2}+2\,t-x}{{x}^{2}{t}^{3}}\sqrt {1-2\, \left( x+{ x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{2}{t}^{2}}\int \!{\frac {10\,{x}^{4}{t}^{3}+12\,{x}^{2}{t}^{3}-6\,{t}^{3}-6\,{x}^{3}{t}^{2}-6\, {t}^{2}x-3\,t-3\,t{x}^{2}+2\,x+x {{}_{2}F_{1}\!\left(-3/2,-1/2;1;16\,{t}^{2}\right)}+3\, \left( t{x}^{2} +t-x \right) {{}_{2}F_{1}\!\left(-1/2,-1/2;2;16\,{t}^{2}\right)}}{ \left( 2\,t{x}^{2}+2\,t-x \right) ^{2}} \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{2}{t}^{2} \right) ^{-3/2}} \,{\rm d}t} 0y\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle 1/2\,{\frac { \left( y+1 \right) ^{4}{t}^{2}-{y}^{2}}{y{t}^{4}}\sqrt {1 -4\,{\frac { \left( y+1 \right) ^{2}{t}^{2}}{y}}}\int \!{\frac {t \left( 4\, \left( y+3 \right) \left( y+1 \right) ^{4}{t}^{4}+ \left( -2\,{y}^{3}+6\,{y}^{2}+6\,y+6 \right) {t}^{2}- \left( y+1 \right) y+ \left( y+1 \right) \left( y {{}_{2}F_{1}\!\left(-3/2,-1/2;1;16\,{t}^{2}\right)}-6\, \left( y+1 \right) ^{2}{t}^{2}{{}_{2}F_{1}\!\left(-1/2,-1/2;2;16\,{t}^{2}\right)} \right) \right) }{ \left( \left( y+1 \right) ^{4}{t}^{2}-{y}^{2} \right) ^{2}} \left( 1-4\,{\frac { \left( y+1 \right) ^{2}{t}^{2}}{y}} \right) ^{-3/2}}\,{\rm d}t}$
xy$\displaystyle {{}_{2}F_{1}\!\left(1/2,1/2;1;16\,{t}^{2}\right)}$$\displaystyle {\frac {1}{xy-t \left( {x}^{2}y+{x}^{2}+{y}^{2}+y \right) } \left( xy-1 /2\,{\frac {2\,t{x}^{2}+2\,t-x}{{t}^{2}}\sqrt {1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{2}{t}^{2}}\int \!{\frac {10\,{x }^{4}{t}^{3}+12\,{x}^{2}{t}^{3}-6\,{t}^{3}-6\,{x}^{3}{t}^{2}-6\,{t}^{2} x-3\,t-3\,t{x}^{2}+2\,x+x {{}_{2}F_{1}\!\left(-3/2,-1/2;1;16\,{t}^{2}\right)}+3\, \left( t{x}^{2} +t-x \right) {{}_{2}F_{1}\!\left(-1/2,-1/2;2;16\,{t}^{2}\right)}}{ \left( 2\,t{x}^{2}+2\,t-x \right) ^{2}} \left( 1-2\, \left( x+{x}^{-1} \right) t+ \left( x-{x}^{-1} \right) ^{2}{t}^{2} \right) ^{-3/2}} \,{\rm d}t}-1/2\,{\frac { \left( y+1 \right) \left( \left( y+1 \right) ^{4}{t}^{2}-{y}^{2} \right) }{{t}^{3}}\sqrt {1-4\,{\frac { \left( y+1 \right) ^{2}{t}^{2}}{y}}}\int \!{\frac {t \left( 4\, \left( y+3 \right) \left( y+1 \right) ^{4}{t}^{4}+ \left( -2\,{y}^{3} +6\,{y}^{2}+6\,y+6 \right) {t}^{2}- \left( y+1 \right) y+ \left( y+1 \right) \left( y{{}_{2}F_{1}\!\left(-3/2,-1/2;1;16\,{t}^{2}\right)}-6 \, \left( y+1 \right) ^{2}{t}^{2} {{}_{2}F_{1}\!\left(-1/2,-1/2;2;16\,{t}^{2}\right)} \right) \right) }{ \left( \left( y+1 \right) ^{4}{t}^{2}-{y}^{2} \right) ^{2}} \left( 1- 4\,{\frac { \left( y+1 \right) ^{2}{t}^{2}}{y}} \right) ^{-3/2}} \,{\rm d}t} \right) }$