Integrand size = 28, antiderivative size = 725 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3} \] Output:
b*(f*x+e)^2/a^2/d+(f*x+e)^2*arctanh(exp(d*x+c))/a/d-2*b^2*(f*x+e)^2*arctan h(exp(d*x+c))/a^3/d-f^2*arctanh(cosh(d*x+c))/a/d^3+b*(f*x+e)^2*coth(d*x+c) /a^2/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)/a /d-b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2 )/d+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/ 2)/d-2*b*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d^2+f*(f*x+e)*polylog(2,-exp(d *x+c))/a/d^2-2*b^2*f*(f*x+e)*polylog(2,-exp(d*x+c))/a^3/d^2-f*(f*x+e)*poly log(2,exp(d*x+c))/a/d^2+2*b^2*f*(f*x+e)*polylog(2,exp(d*x+c))/a^3/d^2-2*b^ 3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/ 2)/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a ^2+b^2)^(1/2)/d^2-b*f^2*polylog(2,exp(2*d*x+2*c))/a^2/d^3-f^2*polylog(3,-e xp(d*x+c))/a/d^3+2*b^2*f^2*polylog(3,-exp(d*x+c))/a^3/d^3+f^2*polylog(3,ex p(d*x+c))/a/d^3-2*b^2*f^2*polylog(3,exp(d*x+c))/a^3/d^3+2*b^3*f^2*polylog( 3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^3-2*b^3*f^2*pol ylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(1/2)/d^3
Leaf count is larger than twice the leaf count of optimal. \(1532\) vs. \(2(725)=1450\).
Time = 8.07 (sec) , antiderivative size = 1532, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
(8*a*b*d^2*e*E^(2*c)*f*x + 4*a*b*d^2*E^(2*c)*f^2*x^2 - 2*a^2*d^2*e^2*ArcTa nh[E^(c + d*x)] + 4*b^2*d^2*e^2*ArcTanh[E^(c + d*x)] + 2*a^2*d^2*e^2*E^(2* c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] + 4*a ^2*f^2*ArcTanh[E^(c + d*x)] - 4*a^2*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] + 2*a ^2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 4*b^2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 4*b^2*d^2*e*E^(2*c)*f*x*Lo g[1 - E^(c + d*x)] + a^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 2*b^2*d^2*f^2* x^2*Log[1 - E^(c + d*x)] - a^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] - 2*a^2*d^2*e*f*x*Log[1 + E ^(c + d*x)] + 4*b^2*d^2*e*f*x*Log[1 + E^(c + d*x)] + 2*a^2*d^2*e*E^(2*c)*f *x*Log[1 + E^(c + d*x)] - 4*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - a ^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 2*b^2*d^2*f^2*x^2*Log[1 + E^(c + d*x )] + a^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] - 2*b^2*d^2*E^(2*c)*f^2* x^2*Log[1 + E^(c + d*x)] + 4*a*b*d*e*f*Log[1 - E^(2*(c + d*x))] - 4*a*b*d* e*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + 4*a*b*d*f^2*x*Log[1 - E^(2*(c + d*x ))] - 4*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(2*(c + d*x))] + 2*(a^2 - 2*b^2)*d*( -1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -E^(c + d*x)] - 2*(a^2 - 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + 2*a*b*f^2*PolyLog[2, E^( 2*(c + d*x))] - 2*a*b*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] + 2*a^2*f^2* PolyLog[3, -E^(c + d*x)] - 4*b^2*f^2*PolyLog[3, -E^(c + d*x)] - 2*a^2*E...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x)^2 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \int (e+f x)^2 \csc (i c+i d x)^3dx}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {f^2 \int -i \text {csch}(c+d x)dx}{d^2}+\frac {1}{2} \int -i (e+f x)^2 \text {csch}(c+d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {i f^2 \int \text {csch}(c+d x)dx}{d^2}-\frac {1}{2} i \int (e+f x)^2 \text {csch}(c+d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {i f^2 \int i \csc (i c+i d x)dx}{d^2}-\frac {1}{2} i \int i (e+f x)^2 \csc (i c+i d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (-\frac {f^2 \int \csc (i c+i d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^2 \csc (i c+i d x)dx-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \int (e+f x)^2 \csc (i c+i d x)dx-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \text {csch}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -(e+f x)^2 \csc (i c+i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\int (e+f x)^2 \csc (i c+i d x)^2dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \coth (c+d x)dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int (e+f x)^2 \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {\int i (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {b \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {b \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3803 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (-\frac {2 b \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-\frac {b \left (-\frac {b \left (\frac {2 b \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
Input:
Int[((e + f*x)^2*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 10341 vs. \(2 (676) = 1352\).
Time = 0.35 (sec) , antiderivative size = 10341, normalized size of antiderivative = 14.26 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**2*csch(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-1/2*e^2*(2*b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^3*d) - 2*(a*e^(-d*x - c) + 2*b *e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a ^2*e^(-4*d*x - 4*c) - a^2)*d) - (a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d ) + (a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d)) - (2*b*d*f^2*x^2 + 4*b*d* e*f*x + (a*d*f^2*x^2*e^(3*c) + 2*a*e*f*e^(3*c) + 2*(d*e*f + f^2)*a*x*e^(3* c))*e^(3*d*x) - 2*(b*d*f^2*x^2*e^(2*c) + 2*b*d*e*f*x*e^(2*c))*e^(2*d*x) + (a*d*f^2*x^2*e^c - 2*a*e*f*e^c + 2*(d*e*f - f^2)*a*x*e^c)*e^(d*x))/(a^2*d^ 2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) + (2*b*d*e*f + a* f^2)*x/(a^2*d^2) + (2*b*d*e*f - a*f^2)*x/(a^2*d^2) - (2*b*d*e*f + a*f^2)*l og(e^(d*x + c) + 1)/(a^2*d^3) - (2*b*d*e*f - a*f^2)*log(e^(d*x + c) - 1)/( a^2*d^3) + 1/2*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*(a^2*f^2 - 2*b^2*f^2)/(a^3*d^3) - 1/2*(d^2*x^ 2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 - 2*b^2*f^2)/(a^3*d^3) + (a^2*d*e*f - 2*b^2*d*e*f - 2*a*b*f ^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) - (a^2*d*e* f - 2*b^2*d*e*f + 2*a*b*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c )))/(a^3*d^3) + 1/6*((a^2*f^2 - 2*b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - 2*b^2* d*e*f + 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/6*((a^2*f^2 - 2*b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - 2*b^2*d*e*f - 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - integra...
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:
int((e + f*x)^2/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
Output:
int((e + f*x)^2/(sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
( - 4*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqr t(a**2 + b**2))*b**4*d**2*e**2*i + 8*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*at an((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**4*d**2*e**2*i - 4*sqrt(a **2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*b**4*d**2*e** 2*i - 64*e**(7*c + 4*d*x)*int((e**(3*d*x)*x**2)/(e**(8*c + 8*d*x)*b + 2*e* *(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x) *a + b),x)*a**6*d**3*f**2 - 64*e**(7*c + 4*d*x)*int((e**(3*d*x)*x**2)/(e** (8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d *x)*b - 2*e**(c + d*x)*a + b),x)*a**4*b**2*d**3*f**2 - 128*e**(7*c + 4*d*x )*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6* c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3 *d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**6*d**3*e*f - 128*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5*d*x)*a + 6*e**(4*c + 4*d*x )*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b), x)*a**4*b**2*d**3*e*f + 32*e**(6*c + 4*d*x)*int((e**(2*d*x)*x**2)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7*d*x)*a - 4*e**(6*c + 6*d*x)*b - 6*e**(5*c + 5* d*x)*a + 6*e**(4*c + 4*d*x)*b + 6*e**(3*c + 3*d*x)*a - 4*e**(2*c + 2*d*...