> printlevel:=100:
> int((sin(x)/x)^5,x);
{--> enter sin, args = x
sin(x) := sin(x)
<-- exit sin (now at top level) = sin(x)}
#01
{--> enter int, args = sin(x)^5/x^5, x
#02
{--> enter int/int, args = [sin(x)^5/x^5, x], 10, _EnvCauchyPrincipalValue
#03
{--> enter int/check_syntax, args = [sin(x)^5/x^5, x]
answer := FAIL
n := 2
5
sin(x)
ff := -------
5
x
llimits := x
5
sin(x)
answer := [-------, x]
5
x
opts := []
opts := []
opts := [formula]
opt := formula
5
sin(x)
answer := [-------, x, [formula]]
5
x
5
sin(x)
[-------, x, [formula]]
5
x
<-- exit int/check_syntax (now in int/int) = [sin(x)^5/x^5, x, [formula]]}
5
sin(x)
answer := [-------, x, [formula]]
5
x
#04
{--> enter int/tools/rename, args = [sin(x)^5/x^5, x, [formula]]
known_bound_fns := {int, sum, product, limit, Int, Limit, Sum, Product}
affected_vars := []
forw_subs := []
{--> enter unknown, args = x
false
<-- exit unknown (now in int/tools/rename) = false}
{--> enter unknown, args = formula
false
<-- exit unknown (now in int/tools/rename) = false}
symb := {}
forw_subs := []
back_subs := []
value remembered (in int/tools/rename): sin(x) -> sin(x)
5
sin(x)
[[-------, x, [formula]], []]
5
x
<-- exit int/tools/rename (now in int/int) = [[sin(x)^5/x^5, x, [formula]], []]}
5
sin(x)
answer := [[-------, x, [formula]], []]
5
x
lt := []
5
sin(x)
answer := [-------, x, [formula]]
5
x
#05
{--> enter int/indefinite, args = sin(x)^5/x^5, x, [formula]
opts := formula
{--> enter int/indef0, args = sin(x)^5/x^5, x
5
sin(_X)
f := --------
5
_X
#06
{--> enter int/indef1, args = sin(_X)^5/_X^5
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
5
sin(_X)
answer := int/itable[--------]
5
_X
#07
answer := int/indef1/NOVAL
#08
{--> enter remove, args = has, sin(_X)^5/_X^5, _X
{--> enter unknown, args = sin(_X)^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
1
<-- exit remove (now in int/indef1) = 1}
#09
c := 1
5
sin(_X)
g := --------
5
_X
#10
{--> enter type/radfun, args = sin(_X)^5/_X^5, anything, _X
v := {_X}
r := sin(_X)
<-- exit type/radfun (now in int/indef1) = false}
#11
{--> enter int/indef2, args = sin(_X)^5/_X^5
{--> enter unknown, args = _X
_X
<-- exit unknown (now in int/indef2) = _X}
{--> enter unknown, args = sin(_X)
sin(_X)
<-- exit unknown (now in int/indef2) = sin(_X)}
inds := {_X, sin(_X)}
{--> enter unknown, args = _X
<-- exit unknown (now in int/indef2) = }
{--> enter unknown, args = sin(_X)
sin(_X)
<-- exit unknown (now in int/indef2) = sin(_X)}
fcns := {sin(_X)}
algs := {}
bases := {}
{--> enter unknown, args = sin(_X)
_X
<-- exit unknown (now in int/indef2) = _X}
fargs := {_X}
{--> enter unknown, args = sin(_X)
sin
<-- exit unknown (now in int/indef2) = sin}
fnames := {sin}
{--> enter int/denoms, args = sin(_X)^5/_X^5
{--> enter sin, args = _X
sin(_X) := sin(_X)
#12
<-- exit sin (now in int/denoms) = sin(_X)}
{--> enter int/denoms, args = sin(_X)^5
value remembered (in int/denoms): sin(_X) -> sin(_X)
{--> enter int/denoms, args = sin(_X)
value remembered (in int/denoms): sin(_X) -> sin(_X)
{--> enter int/denoms, args = _X
{}
<-- exit int/denoms (now in int/denoms) = {}}
{}
<-- exit int/denoms (now in int/denoms) = {}}
value remembered (in int/denoms): sin(_X) -> sin(_X)
{--> enter int/denoms, args = 5
{}
<-- exit int/denoms (now in int/denoms) = {}}
{}
<-- exit int/denoms (now in int/denoms) = {}}
value remembered (in int/denoms): sin(_X) -> sin(_X)
{--> enter int/denoms, args = 1/_X^5
value remembered (in int/denoms): int/denoms(_X) -> {}
{_X}
<-- exit int/denoms (now in int/denoms) = {_X}}
{_X}
<-- exit int/denoms (now in int/indef2) = {_X}}
denoms := {_X}
dummies := {}
#13
{--> enter remove, args = depends, {_X, sin(_X)}, {}
{--> enter unknown, args = _X, {}
{--> enter depends, args = _X, {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = sin(_X), {}
{--> enter depends, args = sin(_X), {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{_X, sin(_X)}
<-- exit remove (now in int/indef2) = {_X, sin(_X)}}
ddset := {_X, sin(_X)}
z := _X
dz := 1
z := sin(_X)
{--> enter diff/sin, args = _X, _X
{--> enter cos, args = _X
cos(_X) := cos(_X)
<-- exit cos (now in diff/sin) = cos(_X)}
cos(_X)
<-- exit diff/sin (now in int/indef2) = cos(_X)}
dz := cos(_X)
5
_Y
t := -----------
5
_X cos(_X)
#14
{--> enter int/trigon, args = sin(_X)^5/_X^5
{--> enter type/radext, args = sin(_X)^5/_X^5
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = sin(_X)^5
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = sin(_X)
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = _X
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = 5
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = 1/_X^5
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = -5
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = sin(_X)^5/_X^5, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = sin(_X)^5, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = sin(_X), constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = _X, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = 5, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = 1/_X^5, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter type/radext, args = -5, constant
false
<-- exit type/radext (now in int/trigon) = false}
{--> enter unknown, args = sin(_X)
sin(_X)
<-- exit unknown (now in int/trigon) = sin(_X)}
{--> enter unknown, args = sin(_X)
<-- exit unknown (now in int/trigon) = }
ff := {}
5
sin(_X)
ff := --------
5
_X
5
Y
ffx := ---
5
_X
dx := -5
ds := 5
dc := 0
#15
{--> enter int/xsincos, args = -5, 5, 0
{--> enter trig/tfourier, args = 5, 0, _X
{--> enter tfourier/sin, args = 5
x := _X
x := _X
#16
{--> enter sum, args = (-1)^k*binomial(5,k)*sin((5-2*k)*x), k = 0 .. 2
xx := k
has_diffs := false
{--> enter type/RootOf, args = 0 .. 2
false
<-- exit type/RootOf (now in sum) = false}
xx := k
a := 0
b := 3
dab := 3
dxx := FAIL
{--> enter binomial, args = 5, 0
1
<-- exit binomial (now in sum) = 1}
{--> enter sin, args = 5*_X
{--> enter csgn, args = 5
z := 5
<-- exit csgn (now in sin) = 1}
sin(5 _X) := sin(5 _X)
<-- exit sin (now in sum) = sin(5*_X)}
{--> enter binomial, args = 5, 1
c := 1
c := 5
binomial/a := 5
binomial/b := 1
binomial/c := 5
<-- exit binomial (now in sum) = 5}
{--> enter sin, args = 3*_X
{--> enter csgn, args = 3
z := 3
<-- exit csgn (now in sin) = 1}
sin(3 _X) := sin(3 _X)
<-- exit sin (now in sum) = sin(3*_X)}
{--> enter binomial, args = 5, 2
c := 5
c := 10
binomial/b := 2
binomial/c := 10
<-- exit binomial (now in sum) = 10}
value remembered (in sum): sin(_X) -> sin(_X)
sin(5 _X) - 5 sin(3 _X) + 10 sin(_X)
<-- exit sum (now in tfourier/sin) = sin(5*_X)-5*sin(3*_X)+10*sin(_X)}
2 sin(5 _X) - 10 sin(3 _X) + 20 sin(_X)
<-- exit tfourier/sin (now in trig/tfourier) = 2*sin(5*_X)-10*sin(3*_X)+20*sin(_X)}
1/16 sin(5 _X) - 5/16 sin(3 _X) + 5/8 sin(_X)
<-- exit trig/tfourier (now in int/xsincos) = 1/16*sin(5*_X)-5/16*sin(3*_X)+5/8*sin(_X)}
#17
{--> enter int/indef1, args = 1/16/_X^5*sin(5*_X)-5/16/_X^5*sin(3*_X)+5/8/_X^5*sin(_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(5 _X) sin(3 _X) sin(_X)
answer := int/itable[1/16 --------- - 5/16 --------- + 5/8 -------]
5 5 5
_X _X _X
answer := int/indef1/NOVAL
#18
{--> enter int/indef1, args = 1/16/_X^5*sin(5*_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(5 _X)
answer := int/itable[1/16 ---------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 1/16/_X^5*sin(5*_X), _X
{--> enter unknown, args = 1/16, _X
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = sin(5*_X), _X
false
<-- exit unknown (now in remove) = false}
1/16
<-- exit remove (now in int/indef1) = 1/16}
c := 1/16
sin(5 _X)
g := ---------
5
_X
#19
{--> enter int/indef1, args = 1/_X^5*sin(5*_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(5 _X)
answer := int/itable[---------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 1/_X^5*sin(5*_X), _X
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = sin(5*_X), _X
false
<-- exit unknown (now in remove) = false}
1
<-- exit remove (now in int/indef1) = 1}
c := 1
sin(5 _X)
g := ---------
5
_X
{--> enter type/radfun, args = 1/_X^5*sin(5*_X), anything, _X
v := {_X}
r := sin(5 _X)
<-- exit type/radfun (now in int/indef1) = false}
#20
{--> enter int/indef2, args = 1/_X^5*sin(5*_X)
{--> enter unknown, args = sin(5*_X)
sin(5 _X)
<-- exit unknown (now in int/indef2) = sin(5*_X)}
{--> enter unknown, args = _X
_X
<-- exit unknown (now in int/indef2) = _X}
inds := {sin(5 _X), _X}
{--> enter unknown, args = sin(5*_X)
sin(5 _X)
<-- exit unknown (now in int/indef2) = sin(5*_X)}
{--> enter unknown, args = _X
<-- exit unknown (now in int/indef2) = }
fcns := {sin(5 _X)}
algs := {}
bases := {}
{--> enter unknown, args = sin(5*_X)
5 _X
<-- exit unknown (now in int/indef2) = 5*_X}
fargs := {5 _X}
{--> enter unknown, args = sin(5*_X)
sin
<-- exit unknown (now in int/indef2) = sin}
fnames := {sin}
{--> enter int/denoms, args = 1/_X^5*sin(5*_X)
value remembered (in int/denoms): sin(5*_X) -> sin(5*_X)
value remembered (in int/denoms): int/denoms(1/_X^5) -> {_X}
value remembered (in int/denoms): sin(5*_X) -> sin(5*_X)
{--> enter int/denoms, args = sin(5*_X)
{--> enter int/denoms, args = 5*_X
<-- exit int/denoms (now in int/denoms) = {}}
{}
<-- exit int/denoms (now in int/denoms) = {}}
{_X}
<-- exit int/denoms (now in int/indef2) = {_X}}
denoms := {_X}
dummies := {}
{--> enter remove, args = depends, {sin(5*_X), 5*_X, _X}, {}
{--> enter unknown, args = sin(5*_X), {}
{--> enter depends, args = sin(5*_X), {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = 5*_X, {}
{--> enter depends, args = 5*_X, {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = _X, {}
{--> enter depends, args = _X, {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{sin(5 _X), 5 _X, _X}
<-- exit remove (now in int/indef2) = {sin(5*_X), 5*_X, _X}}
ddset := {sin(5 _X), 5 _X, _X}
z := sin(5 _X)
{--> enter diff/sin, args = 5*_X, _X
{--> enter cos, args = 5*_X
{--> enter csgn, args = 5
<-- exit csgn (now in cos) = 1}
cos(5 _X) := cos(5 _X)
<-- exit cos (now in diff/sin) = cos(5*_X)}
5 cos(5 _X)
<-- exit diff/sin (now in int/indef2) = 5*cos(5*_X)}
dz := 5 cos(5 _X)
_Y
t := 1/5 -------------
5
_X cos(5 _X)
z := 5 _X
dz := 5
z := _X
dz := 1
{--> enter type/indexedfun, args = sin(5*_X), _X
false
<-- exit type/indexedfun (now in int/indef2) = false}
#21
{--> enter int/subs, args = 5*_X, a0, a1, t
{--> enter ispoly, args = 5*_X, linear, _X, a0, a1
<-- exit ispoly (now in int/subs) = true}
a0 := 5 _X
a1 := 1/5 _X
t := _X
<-- exit int/subs (now in int/indef2) = true}
sin(_X)
t := 625 -------
5
_X
#22
{--> enter int/indef1, args = 625/_X^5*sin(_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(_X)
answer := int/itable[625 -------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 625/_X^5*sin(_X), _X
625
<-- exit remove (now in int/indef1) = 625}
c := 625
sin(_X)
g := -------
5
_X
#23
{--> enter int/indef1, args = 1/_X^5*sin(_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(_X)
answer := int/itable[-------]
5
_X
#24
sin(_X) cos(_X) sin(_X) cos(_X)
- 1/4 ------- - 1/12 ------- + 1/24 ------- + 1/24 ------- + 1/24 Si(_X)
4 3 2 _X
_X _X _X
<-- exit int/indef1 (now in int/indef1) = -1/4*sin(_X)/_X^4-1/12*cos(_X)/_X^3+1/24*sin(_X)/_X^2+1/24*cos(_X)/_X+1/24*Si(_X)}
sin(_X) cos(_X) sin(_X) cos(_X)
r := - 1/4 ------- - 1/12 ------- + 1/24 ------- + 1/24 -------
4 3 2 _X
_X _X _X
+ 1/24 Si(_X)
answer :=
sin(_X) 625 cos(_X) 625 sin(_X) 625 cos(_X) 625
- 625/4 ------- - --- ------- + --- ------- + --- ------- + --- Si(_X)
4 12 3 24 2 24 _X 24
_X _X _X
sin(_X) 625 cos(_X) 625 sin(_X) 625 cos(_X) 625
- 625/4 ------- - --- ------- + --- ------- + --- ------- + --- Si(_X)
4 12 3 24 2 24 _X 24
_X _X _X
<-- exit int/indef1 (now in int/indef2) = -625/4*sin(_X)/_X^4-625/12*cos(_X)/_X^3+625/24*sin(_X)/_X^2+625/24*cos(_X)/_X+625/24*Si(_X)}
<-- exit int/indef2 (now in int/indef1) = -1/4*sin(5*_X)/_X^4-5/12*cos(5*_X)/_X^3+25/24*sin(5*_X)/_X^2+125/24*cos(5*_X)/_X+625/24*Si(5*_X)}
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
answer := - 1/4 --------- - 5/12 --------- + -- --------- + --- ---------
4 3 24 2 24 _X
_X _X _X
625
+ --- Si(5 _X)
24
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
- 1/4 --------- - 5/12 --------- + -- --------- + --- ---------
4 3 24 2 24 _X
_X _X _X
625
+ --- Si(5 _X)
24
<-- exit int/indef1 (now in int/indef1) = -1/4*sin(5*_X)/_X^4-5/12*cos(5*_X)/_X^3+25/24*sin(5*_X)/_X^2+125/24*cos(5*_X)/_X+625/24*Si(5*_X)}
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
r := - 1/4 --------- - 5/12 --------- + -- --------- + --- ---------
4 3 24 2 24 _X
_X _X _X
625
+ --- Si(5 _X)
24
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
answer := - 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625
+ --- Si(5 _X)
384
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
- 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625
+ --- Si(5 _X)
384
<-- exit int/indef1 (now in int/indef1) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)}
#25
{--> enter int/indef1, args = -5/16/_X^5*sin(3*_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(3 _X)
answer := int/itable[- 5/16 ---------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, -5/16/_X^5*sin(3*_X), _X
{--> enter unknown, args = -5/16, _X
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = sin(3*_X), _X
false
<-- exit unknown (now in remove) = false}
-5
--
16
<-- exit remove (now in int/indef1) = -5/16}
-5
c := --
16
sin(3 _X)
g := ---------
5
_X
{--> enter int/indef1, args = 1/_X^5*sin(3*_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(3 _X)
answer := int/itable[---------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 1/_X^5*sin(3*_X), _X
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = sin(3*_X), _X
false
<-- exit unknown (now in remove) = false}
1
<-- exit remove (now in int/indef1) = 1}
c := 1
sin(3 _X)
g := ---------
5
_X
{--> enter type/radfun, args = 1/_X^5*sin(3*_X), anything, _X
v := {_X}
r := sin(3 _X)
<-- exit type/radfun (now in int/indef1) = false}
{--> enter int/indef2, args = 1/_X^5*sin(3*_X)
{--> enter unknown, args = sin(3*_X)
sin(3 _X)
<-- exit unknown (now in int/indef2) = sin(3*_X)}
{--> enter unknown, args = _X
_X
<-- exit unknown (now in int/indef2) = _X}
inds := {sin(3 _X), _X}
{--> enter unknown, args = sin(3*_X)
sin(3 _X)
<-- exit unknown (now in int/indef2) = sin(3*_X)}
{--> enter unknown, args = _X
<-- exit unknown (now in int/indef2) = }
fcns := {sin(3 _X)}
algs := {}
bases := {}
{--> enter unknown, args = sin(3*_X)
3 _X
<-- exit unknown (now in int/indef2) = 3*_X}
fargs := {3 _X}
{--> enter unknown, args = sin(3*_X)
sin
<-- exit unknown (now in int/indef2) = sin}
fnames := {sin}
{--> enter int/denoms, args = 1/_X^5*sin(3*_X)
value remembered (in int/denoms): sin(3*_X) -> sin(3*_X)
value remembered (in int/denoms): int/denoms(1/_X^5) -> {_X}
value remembered (in int/denoms): sin(3*_X) -> sin(3*_X)
{--> enter int/denoms, args = sin(3*_X)
{--> enter int/denoms, args = 3*_X
<-- exit int/denoms (now in int/denoms) = {}}
{}
<-- exit int/denoms (now in int/denoms) = {}}
{_X}
<-- exit int/denoms (now in int/indef2) = {_X}}
denoms := {_X}
dummies := {}
{--> enter remove, args = depends, {sin(3*_X), 3*_X, _X}, {}
{--> enter unknown, args = sin(3*_X), {}
{--> enter depends, args = sin(3*_X), {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = 3*_X, {}
{--> enter depends, args = 3*_X, {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = _X, {}
{--> enter depends, args = _X, {}
<-- exit depends (now in unknown) = false}
true
<-- exit unknown (now in remove) = true}
{sin(3 _X), 3 _X, _X}
<-- exit remove (now in int/indef2) = {sin(3*_X), 3*_X, _X}}
ddset := {sin(3 _X), 3 _X, _X}
z := sin(3 _X)
{--> enter diff/sin, args = 3*_X, _X
{--> enter cos, args = 3*_X
{--> enter csgn, args = 3
<-- exit csgn (now in cos) = 1}
cos(3 _X) := cos(3 _X)
<-- exit cos (now in diff/sin) = cos(3*_X)}
3 cos(3 _X)
<-- exit diff/sin (now in int/indef2) = 3*cos(3*_X)}
dz := 3 cos(3 _X)
_Y
t := 1/3 -------------
5
_X cos(3 _X)
z := 3 _X
dz := 3
z := _X
dz := 1
{--> enter type/indexedfun, args = sin(3*_X), _X
false
<-- exit type/indexedfun (now in int/indef2) = false}
{--> enter int/subs, args = 3*_X, a0, a1, t
{--> enter ispoly, args = 3*_X, linear, _X, a0, a1
<-- exit ispoly (now in int/subs) = true}
a0 := 3 _X
a1 := 1/3 _X
t := _X
<-- exit int/subs (now in int/indef2) = true}
sin(_X)
t := 81 -------
5
_X
{--> enter int/indef1, args = 81/_X^5*sin(_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(_X)
answer := int/itable[81 -------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 81/_X^5*sin(_X), _X
81
<-- exit remove (now in int/indef1) = 81}
c := 81
sin(_X)
g := -------
5
_X
sin(_X) cos(_X) sin(_X) cos(_X)
r := - 1/4 ------- - 1/12 ------- + 1/24 ------- + 1/24 -------
4 3 2 _X
_X _X _X
+ 1/24 Si(_X)
sin(_X) cos(_X) sin(_X) cos(_X)
answer := - 81/4 ------- - 27/4 ------- + 27/8 ------- + 27/8 -------
4 3 2 _X
_X _X _X
+ 27/8 Si(_X)
sin(_X) cos(_X) sin(_X) cos(_X)
- 81/4 ------- - 27/4 ------- + 27/8 ------- + 27/8 ------- + 27/8 Si(_X)
4 3 2 _X
_X _X _X
<-- exit int/indef1 (now in int/indef2) = -81/4*sin(_X)/_X^4-27/4*cos(_X)/_X^3+27/8*sin(_X)/_X^2+27/8*cos(_X)/_X+27/8*Si(_X)}
<-- exit int/indef2 (now in int/indef1) = -1/4*sin(3*_X)/_X^4-1/4*cos(3*_X)/_X^3+3/8*sin(3*_X)/_X^2+9/8*cos(3*_X)/_X+27/8*Si(3*_X)}
sin(3 _X) cos(3 _X) sin(3 _X) cos(3 _X)
answer := - 1/4 --------- - 1/4 --------- + 3/8 --------- + 9/8 ---------
4 3 2 _X
_X _X _X
+ 27/8 Si(3 _X)
sin(3 _X) cos(3 _X) sin(3 _X) cos(3 _X)
- 1/4 --------- - 1/4 --------- + 3/8 --------- + 9/8 ---------
4 3 2 _X
_X _X _X
+ 27/8 Si(3 _X)
<-- exit int/indef1 (now in int/indef1) = -1/4*sin(3*_X)/_X^4-1/4*cos(3*_X)/_X^3+3/8*sin(3*_X)/_X^2+9/8*cos(3*_X)/_X+27/8*Si(3*_X)}
sin(3 _X) cos(3 _X) sin(3 _X) cos(3 _X)
r := - 1/4 --------- - 1/4 --------- + 3/8 --------- + 9/8 ---------
4 3 2 _X
_X _X _X
+ 27/8 Si(3 _X)
sin(3 _X) cos(3 _X) 15 sin(3 _X) 45 cos(3 _X)
answer := 5/64 --------- + 5/64 --------- - --- --------- - --- ---------
4 3 128 2 128 _X
_X _X _X
135
- --- Si(3 _X)
128
sin(3 _X) cos(3 _X) 15 sin(3 _X) 45 cos(3 _X)
5/64 --------- + 5/64 --------- - --- --------- - --- ---------
4 3 128 2 128 _X
_X _X _X
135
- --- Si(3 _X)
128
<-- exit int/indef1 (now in int/indef1) = 5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)}
#26
{--> enter int/indef1, args = 5/8/_X^5*sin(_X)
table([
_X
----------- =
1 + sin(_X)
_X cos(1/2 _X)
-2 ------------------------- + _X + 2 ln(sin(1/2 _X) + cos(1/2 _X))
sin(1/2 _X) + cos(1/2 _X)
cosh(_X) = sinh(_X)
arccosh(_X) = _X arccosh(_X) - sqrt(_X - 1) sqrt(_X + 1)
2
arccot(_X) = arccot(_X) _X + 1/2 ln(_X + 1)
cot(_X) = ln(sin(_X))
1
arccsc(_X) = arccsc(_X) _X + ln(_X + _X sqrt(1 - ---))
2
_X
sqrt(cot(_X)) =
sqrt(cot(_X)) sin(_X) sqrt(2) arccos(sin(_X) - cos(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(sin(_X) + sqrt(2) sqrt(cot(_X)) sin(_X) + cos(_X))
sqrt(coth(_X)) = arctanh(sqrt(coth(_X))) - arctan(sqrt(coth(_X)))
sqrt(tanh(_X)) = arctanh(sqrt(tanh(_X))) - arctan(sqrt(tanh(_X)))
2
arcsin(_X) = _X arcsin(_X) + sqrt(1 - _X )
_X
----------- = -_X cot(1/2 _X) + 2 ln(sin(1/2 _X))
1 - cos(_X)
_X - sin(_X)
------------ = -_X cot(1/2 _X)
1 - cos(_X)
2
arctan(_X) = _X arctan(_X) - 1/2 ln(_X + 1)
exp(_X) = exp(_X)
sec(_X) = ln(sec(_X) + tan(_X))
sinh(_X) = cosh(_X)
_X + sin(_X)
------------ = _X tan(1/2 _X)
1 + cos(_X)
sech(_X) = arctan(sinh(_X))
sin(ln(_X)) = - 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
csch(_X) = ln(tanh(1/2 _X))
tan(_X) = -ln(cos(_X))
sin(_X) = -cos(_X)
cos(_X) = sin(_X)
2
arctanh(_X) = _X arctanh(_X) + 1/2 ln(1 - _X )
ln(_X) = _X ln(_X) - _X
coth(_X) = ln(sinh(_X))
_X
----------- = _X tan(1/2 _X) + 2 ln(cos(1/2 _X))
1 + cos(_X)
_X
----------- =
1 - sin(_X)
_X cos(1/2 _X)
2 -------------------------- - _X + 2 ln(-sin(1/2 _X) + cos(1/2 _X))
-sin(1/2 _X) + cos(1/2 _X)
1
arcsec(_X) = _X arcsec(_X) - ln(_X + _X sqrt(1 - ---))
2
_X
cos(ln(_X)) = 1/2 cos(ln(_X)) _X + 1/2 sin(ln(_X)) _X
1 1
arcsech(_X) = _X arcsech(_X) - arctan(sqrt(---- - 1) sqrt(---- + 1))
_X _X
1
arccsch(_X) = _X arccsch(_X) + ln(_X + _X sqrt(1 + ---))
2
_X
csc(_X) = -ln(csc(_X) + cot(_X))
sqrt(tan(_X)) =
sqrt(tan(_X)) cos(_X) sqrt(2) arccos(cos(_X) - sin(_X))
1/2 -------------------------------------------------------
sqrt(cos(_X) sin(_X))
- 1/2 sqrt(2) ln(cos(_X) + sqrt(2) sqrt(tan(_X)) cos(_X) + sin(_X))
tanh(_X) = ln(cosh(_X))
2
arcsinh(_X) = _X arcsinh(_X) - sqrt(_X + 1)
2
arccos(_X) = _X arccos(_X) - sqrt(1 - _X )
2
arccoth(_X) = _X arccoth(_X) + 1/2 ln(_X - 1)
])
sin(_X)
answer := int/itable[5/8 -------]
5
_X
answer := int/indef1/NOVAL
{--> enter remove, args = has, 5/8/_X^5*sin(_X), _X
{--> enter unknown, args = 5/8, _X
true
<-- exit unknown (now in remove) = true}
{--> enter unknown, args = 1/_X^5, _X
false
<-- exit unknown (now in remove) = false}
{--> enter unknown, args = sin(_X), _X
false
<-- exit unknown (now in remove) = false}
5/8
<-- exit remove (now in int/indef1) = 5/8}
c := 5/8
sin(_X)
g := -------
5
_X
value remembered (in int/indef1): int/indef1(1/_X^5*sin(_X)) -> -1/4*sin(_X)/_X^4-1/12*cos(_X)/_X^3+1/24*sin(_X)/_X^2+1/24*cos(_X)/_X+1/24*Si(_X)
sin(_X) cos(_X) sin(_X) cos(_X)
r := - 1/4 ------- - 1/12 ------- + 1/24 ------- + 1/24 -------
4 3 2 _X
_X _X _X
+ 1/24 Si(_X)
sin(_X) cos(_X) sin(_X) cos(_X)
answer := - 5/32 ------- - 5/96 ------- + 5/192 ------- + 5/192 -------
4 3 2 _X
_X _X _X
+ 5/192 Si(_X)
sin(_X) cos(_X) sin(_X) cos(_X)
- 5/32 ------- - 5/96 ------- + 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
4 3 2 _X
_X _X _X
<-- exit int/indef1 (now in int/indef1) = -5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
answer := - 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
- 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
<-- exit int/indef1 (now in int/xsincos) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)+5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)-5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
<-- exit int/xsincos (now in int/trigon) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)+5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)-5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
<-- exit int/trigon (now in int/indef2) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)+5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)-5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
#27
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
r := - 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
<-- exit int/indef2 (now in int/indef1) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)+5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)-5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
answer := - 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
- 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
<-- exit int/indef1 (now in int/indef0) = -1/64*sin(5*_X)/_X^4-5/192*cos(5*_X)/_X^3+25/384*sin(5*_X)/_X^2+125/384*cos(5*_X)/_X+625/384*Si(5*_X)+5/64*sin(3*_X)/_X^4+5/64*cos(3*_X)/_X^3-15/128*sin(3*_X)/_X^2-45/128*cos(3*_X)/_X-135/128*Si(3*_X)-5/32*sin(_X)/_X^4-5/96*cos(_X)/_X^3+5/192*sin(_X)/_X^2+5/192*cos(_X)/_X+5/192*Si(_X)}
sin(5 _X) cos(5 _X) 25 sin(5 _X) 125 cos(5 _X)
answer := - 1/64 --------- - 5/192 --------- + --- --------- + --- ---------
4 3 384 2 384 _X
_X _X _X
625 sin(3 _X) cos(3 _X) 15 sin(3 _X)
+ --- Si(5 _X) + 5/64 --------- + 5/64 --------- - --- ---------
384 4 3 128 2
_X _X _X
45 cos(3 _X) 135 sin(_X) cos(_X)
- --- --------- - --- Si(3 _X) - 5/32 ------- - 5/96 -------
128 _X 128 4 3
_X _X
sin(_X) cos(_X)
+ 5/192 ------- + 5/192 ------- + 5/192 Si(_X)
2 _X
_X
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x) 625
- 1/64 -------- - 5/192 -------- + --- -------- + --- -------- + --- Si(5 x)
4 3 384 2 384 x 384
x x x
sin(3 x) cos(3 x) 15 sin(3 x) 45 cos(3 x)
+ 5/64 -------- + 5/64 -------- - --- -------- - --- --------
4 3 128 2 128 x
x x x
135 sin(x) cos(x) sin(x)
- --- Si(3 x) - 5/32 ------ - 5/96 ------ + 5/192 ------
128 4 3 2
x x x
cos(x)
+ 5/192 ------ + 5/192 Si(x)
x
<-- exit int/indef0 (now in int/indefinite) = -1/64*sin(5*x)/x^4-5/192*cos(5*x)/x^3+25/384*sin(5*x)/x^2+125/384*cos(5*x)/x+625/384*Si(5*x)+5/64*sin(3*x)/x^4+5/64*cos(3*x)/x^3-15/128*sin(3*x)/x^2-45/128*cos(3*x)/x-135/128*Si(3*x)-5/32*sin(x)/x^4-5/96*cos(x)/x^3+5/192*sin(x)/x^2+5/192*cos(x)/x+5/192*Si(x)}
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x)
answer := - 1/64 -------- - 5/192 -------- + --- -------- + --- --------
4 3 384 2 384 x
x x x
625 sin(3 x) cos(3 x) 15 sin(3 x)
+ --- Si(5 x) + 5/64 -------- + 5/64 -------- - --- --------
384 4 3 128 2
x x x
45 cos(3 x) 135 sin(x) cos(x)
- --- -------- - --- Si(3 x) - 5/32 ------ - 5/96 ------
128 x 128 4 3
x x
sin(x) cos(x)
+ 5/192 ------ + 5/192 ------ + 5/192 Si(x)
2 x
x
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x) 625
- 1/64 -------- - 5/192 -------- + --- -------- + --- -------- + --- Si(5 x)
4 3 384 2 384 x 384
x x x
sin(3 x) cos(3 x) 15 sin(3 x) 45 cos(3 x)
+ 5/64 -------- + 5/64 -------- - --- -------- - --- --------
4 3 128 2 128 x
x x x
135 sin(x) cos(x) sin(x)
- --- Si(3 x) - 5/32 ------ - 5/96 ------ + 5/192 ------
128 4 3 2
x x x
cos(x)
+ 5/192 ------ + 5/192 Si(x)
x
<-- exit int/indefinite (now in int/int) = -1/64*sin(5*x)/x^4-5/192*cos(5*x)/x^3+25/384*sin(5*x)/x^2+125/384*cos(5*x)/x+625/384*Si(5*x)+5/64*sin(3*x)/x^4+5/64*cos(3*x)/x^3-15/128*sin(3*x)/x^2-45/128*cos(3*x)/x-135/128*Si(3*x)-5/32*sin(x)/x^4-5/96*cos(x)/x^3+5/192*sin(x)/x^2+5/192*cos(x)/x+5/192*Si(x)}
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x)
answer := - 1/64 -------- - 5/192 -------- + --- -------- + --- --------
4 3 384 2 384 x
x x x
625 sin(3 x) cos(3 x) 15 sin(3 x)
+ --- Si(5 x) + 5/64 -------- + 5/64 -------- - --- --------
384 4 3 128 2
x x x
45 cos(3 x) 135 sin(x) cos(x)
- --- -------- - --- Si(3 x) - 5/32 ------ - 5/96 ------
128 x 128 4 3
x x
sin(x) cos(x)
+ 5/192 ------ + 5/192 ------ + 5/192 Si(x)
2 x
x
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x)
answer := - 1/64 -------- - 5/192 -------- + --- -------- + --- --------
4 3 384 2 384 x
x x x
625 sin(3 x) cos(3 x) 15 sin(3 x)
+ --- Si(5 x) + 5/64 -------- + 5/64 -------- - --- --------
384 4 3 128 2
x x x
45 cos(3 x) 135 sin(x) cos(x)
- --- -------- - --- Si(3 x) - 5/32 ------ - 5/96 ------
128 x 128 4 3
x x
sin(x) cos(x)
+ 5/192 ------ + 5/192 ------ + 5/192 Si(x)
2 x
x
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x) 625
- 1/64 -------- - 5/192 -------- + --- -------- + --- -------- + --- Si(5 x)
4 3 384 2 384 x 384
x x x
sin(3 x) cos(3 x) 15 sin(3 x) 45 cos(3 x)
+ 5/64 -------- + 5/64 -------- - --- -------- - --- --------
4 3 128 2 128 x
x x x
135 sin(x) cos(x) sin(x)
- --- Si(3 x) - 5/32 ------ - 5/96 ------ + 5/192 ------
128 4 3 2
x x x
cos(x)
+ 5/192 ------ + 5/192 Si(x)
x
<-- exit int/int (now in int) = -1/64*sin(5*x)/x^4-5/192*cos(5*x)/x^3+25/384*sin(5*x)/x^2+125/384*cos(5*x)/x+625/384*Si(5*x)+5/64*sin(3*x)/x^4+5/64*cos(3*x)/x^3-15/128*sin(3*x)/x^2-45/128*cos(3*x)/x-135/128*Si(3*x)-5/32*sin(x)/x^4-5/96*cos(x)/x^3+5/192*sin(x)/x^2+5/192*cos(x)/x+5/192*Si(x)}
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x)
answer := - 1/64 -------- - 5/192 -------- + --- -------- + --- --------
4 3 384 2 384 x
x x x
625 sin(3 x) cos(3 x) 15 sin(3 x)
+ --- Si(5 x) + 5/64 -------- + 5/64 -------- - --- --------
384 4 3 128 2
x x x
45 cos(3 x) 135 sin(x) cos(x)
- --- -------- - --- Si(3 x) - 5/32 ------ - 5/96 ------
128 x 128 4 3
x x
sin(x) cos(x)
+ 5/192 ------ + 5/192 ------ + 5/192 Si(x)
2 x
x
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x) 625
- 1/64 -------- - 5/192 -------- + --- -------- + --- -------- + --- Si(5 x)
4 3 384 2 384 x 384
x x x
sin(3 x) cos(3 x) 15 sin(3 x) 45 cos(3 x)
+ 5/64 -------- + 5/64 -------- - --- -------- - --- --------
4 3 128 2 128 x
x x x
135 sin(x) cos(x) sin(x)
- --- Si(3 x) - 5/32 ------ - 5/96 ------ + 5/192 ------
128 4 3 2
x x x
cos(x)
+ 5/192 ------ + 5/192 Si(x)
x
#28
<-- exit int (now at top level) = -1/64*sin(5*x)/x^4-5/192*cos(5*x)/x^3+25/384*sin(5*x)/x^2+125/384*cos(5*x)/x+625/384*Si(5*x)+5/64*sin(3*x)/x^4+5/64*cos(3*x)/x^3-15/128*sin(3*x)/x^2-45/128*cos(3*x)/x-135/128*Si(3*x)-5/32*sin(x)/x^4-5/96*cos(x)/x^3+5/192*sin(x)/x^2+5/192*cos(x)/x+5/192*Si(x)}
sin(5 x) cos(5 x) 25 sin(5 x) 125 cos(5 x) 625
- 1/64 -------- - 5/192 -------- + --- -------- + --- -------- + --- Si(5 x)
4 3 384 2 384 x 384
x x x
sin(3 x) cos(3 x) 15 sin(3 x) 45 cos(3 x)
+ 5/64 -------- + 5/64 -------- - --- -------- - --- --------
4 3 128 2 128 x
x x x
135 sin(x) cos(x) sin(x)
- --- Si(3 x) - 5/32 ------ - 5/96 ------ + 5/192 ------
128 4 3 2
x x x
cos(x)
+ 5/192 ------ + 5/192 Si(x)
x
> printlevel:=1:
> writeto(terminal):