Integrand size = 28, antiderivative size = 725 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3} \]
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-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)*polylo g(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-b*f^2* polylog(2,exp(2*d*x+2*c))/a^2/d^3-f^2*polylog(3,-exp(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,exp(d*x+c))/a/d^3-2*b^2*f^ 2*polylog(3,exp(d*x+c))/a^3/d^3-b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^ 2)^(1/2)))/a^3/d/(a^2+b^2)^(1/2)+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b ^2)^(1/2)))/a^3/d/(a^2+b^2)^(1/2)-2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/ (a-(a^2+b^2)^(1/2)))/a^3/d^2/(a^2+b^2)^(1/2)+2*b^3*f*(f*x+e)*polylog(2,-b* exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^2/(a^2+b^2)^(1/2)+2*b^3*f^2*polylog( 3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^3/(a^2+b^2)^(1/2)-2*b^3*f^2*pol ylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^3/(a^2+b^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1532\) vs. \(2(725)=1450\).
Time = 8.24 (sec) , antiderivative size = 1532, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
(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 |
\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {2 b \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a}+\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}\right )}{a}-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\right )}{a}-\frac {i \left (\frac {1}{2} \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {i f^2 \text {arctanh}(\cosh (c+d x))}{d^3}-\frac {i f (e+f x) \text {csch}(c+d x)}{d^2}-\frac {i (e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}\) |
3.3.49.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* (f_.)*(x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( -I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csch[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 10341 vs. \(2 (676) = 1352\).
Time = 0.40 (sec) , antiderivative size = 10341, normalized size of antiderivative = 14.26 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-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. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]