Integrand size = 26, antiderivative size = 435 \[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{2 a d}-\frac {(e+f x)^2}{2 a f}-\frac {b^2 (e+f x)^2}{2 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^3 f}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2} \]
1/2*f*x/a/d-1/2*(f*x+e)^2/a/f-1/2*b^2*(f*x+e)^2/a^3/f+1/2*(a^2+b^2)*(f*x+e )^2/a^3/f+b*f*arctanh(cosh(d*x+c))/a^2/d^2-1/2*f*coth(d*x+c)/a/d^2-1/2*(f* x+e)*coth(d*x+c)^2/a/d+b*(f*x+e)*csch(d*x+c)/a^2/d+(f*x+e)*ln(1-exp(2*d*x+ 2*c))/a/d+b^2*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^3/d-(a^2+b^2)*(f*x+e)*ln(1+b* exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d-(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/ (a+(a^2+b^2)^(1/2)))/a^3/d+1/2*f*polylog(2,exp(2*d*x+2*c))/a/d^2+1/2*b^2*f *polylog(2,exp(2*d*x+2*c))/a^3/d^2-(a^2+b^2)*f*polylog(2,-b*exp(d*x+c)/(a- (a^2+b^2)^(1/2)))/a^3/d^2-(a^2+b^2)*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2) ^(1/2)))/a^3/d^2
Time = 8.63 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.76 \[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\frac {\left (a^2+b^2\right ) (d e+d f x)^2}{2 f}+\left (-a b f+a^2 (d e+d f x)+b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )+\left (a b f+a^2 (d e+d f x)+b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )-\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a^3 d^2}-\frac {\left (a^2+b^2\right ) \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a^3 d^2}+\frac {(d e-c f+f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )-a f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \]
((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d* x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) + (((a^2 + b^2)*(d*e + d*f*x)^2)/(2*f) + (-(a*b*f) + a^2*(d*e + d*f*x) + b^2*(d*e + d *f*x))*Log[1 - E^(-c - d*x)] + (a*b*f + a^2*(d*e + d*f*x) + b^2*(d*e + d*f *x))*Log[1 + E^(-c - d*x)] - (a^2 + b^2)*f*PolyLog[2, -E^(-c - d*x)] - (a^ 2 + b^2)*f*PolyLog[2, E^(-c - d*x)])/(a^3*d^2) - ((a^2 + b^2)*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[ (a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-( a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^ 2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(- 1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^ 2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(2*a^3*d ^2) + ((d*e - c*f + f*(c + d*x))*Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(-2*b*d*e*Sinh[(c + d*x)/2] - a*f*Sinh[(c + d*x)/2] + 2*b*c*f*S inh[(c + d*x)/2] - 2*b*f*(c + d*x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)
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.
| \(\displaystyle \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6103 |
| \(\displaystyle \frac {\int (e+f x) \coth ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^3dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int i (e+f x) \coth (c+d x)dx+\frac {i f \int -\coth ^2(c+d x)dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int i (e+f x) \coth (c+d x)dx-\frac {i f \int \coth ^2(c+d x)dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int (e+f x) \coth (c+d x)dx-\frac {i f \int \coth ^2(c+d x)dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {i f \int -\tan \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-i \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx+\frac {i f \int \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2dx}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx+\frac {i f \left (\frac {\coth (c+d x)}{d}-\int 1dx\right )}{2 d}+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \cosh (c+d x)dx+\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {i f \int -i \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \int \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \int -i \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx+\frac {i f \int \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle -\frac {b \left (\frac {\frac {f \int \text {csch}(c+d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \int i \csc (i c+i d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {i f \int \csc (i c+i d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \cosh ^2(c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \coth (c+d x)dx+\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\right )}{a}+\frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (-2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )+\frac {i (e+f x) \coth ^2(c+d x)}{2 d}+\frac {i f \left (\frac {\coth (c+d x)}{d}-x\right )}{2 d}+\frac {i (e+f x)^2}{2 f}\right )}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx+\int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\right )}{a}\) |
3.5.88.3.1 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[(((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[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[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[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
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[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Coth[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[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[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S imp[1/a Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Simp[b/ a Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(Coth[c + d*x]^(n - 1)/(a + b*Sin h[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[ n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1097\) vs. \(2(407)=814\).
Time = 1.98 (sec) , antiderivative size = 1098, normalized size of antiderivative = 2.52
-(-2*b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-2*b*d*e*exp(3*d*x+3*c )+2*a*d*e*exp(2*d*x+2*c)+2*b*d*f*x*exp(d*x+c)+a*f*exp(2*d*x+2*c)+2*b*d*e*e xp(d*x+c)-a*f)/a^2/d^2/(exp(2*d*x+2*c)-1)^2-1/d*f/a*ln((b*exp(d*x+c)+(a^2+ b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d*f/a*ln((-b*exp(d*x+c)+(a^2+b^2)^( 1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d*f/a*ln(exp(d*x+c)+1)*x-1/d^2*f/a*ln((b *exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d^2*f/a*ln((-b*exp (d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2*c*f/a*ln(exp(d*x+ c)-1)+1/d^2*c*f/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d^2*b^2/a^3*f*di log(exp(d*x+c))+1/d^2*b^2/a^3*f*dilog(exp(d*x+c)+1)+1/d*b^2/a^3*e*ln(exp(d *x+c)-1)+1/d*b^2/a^3*e*ln(exp(d*x+c)+1)-1/d*b^2/a^3*e*ln(b*exp(2*d*x+2*c)+ 2*a*exp(d*x+c)-b)-1/d^2*b^2/a^3*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/ (-a+(a^2+b^2)^(1/2)))-1/d^2*b^2/a^3*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+ a)/(a+(a^2+b^2)^(1/2)))-1/d^2*b/a^2*f*ln(exp(d*x+c)-1)+1/d^2*b/a^2*f*ln(ex p(d*x+c)+1)-1/d^2*f/a*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2 )^(1/2)))-1/d^2*f/a*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1 /2)))-1/d^2*f/a*dilog(exp(d*x+c))+1/d*e/a*ln(exp(d*x+c)-1)-1/d^2*b^2/a^3*f *ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/d^2*b^2/a^3* f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d*b^2/a^3 *f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d*b^2/a^3* f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d*e/a*...
Leaf count of result is larger than twice the leaf count of optimal. 3547 vs. \(2 (403) = 806\).
Time = 0.31 (sec) , antiderivative size = 3547, normalized size of antiderivative = 8.15 \[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
(2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c)^3 + 2*(a*b*d*f*x + a*b*d*e)*sinh(d* x + c)^3 + a^2*f - (2*a^2*d*f*x + 2*a^2*d*e + a^2*f)*cosh(d*x + c)^2 - (2* a^2*d*f*x + 2*a^2*d*e + a^2*f - 6*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c))*sin h(d*x + c)^2 - 2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c) - ((a^2 + b^2)*f*cosh (d*x + c)^4 + 4*(a^2 + b^2)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)* f*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f*cosh(d*x + c)^2 + 2*(3*(a^2 + b^2)*f*c osh(d*x + c)^2 - (a^2 + b^2)*f)*sinh(d*x + c)^2 + (a^2 + b^2)*f + 4*((a^2 + b^2)*f*cosh(d*x + c)^3 - (a^2 + b^2)*f*cosh(d*x + c))*sinh(d*x + c))*dil og((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c) )*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - ((a^2 + b^2)*f*cosh(d*x + c)^4 + 4*( a^2 + b^2)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)*f*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f*cosh(d*x + c)^2 + 2*(3*(a^2 + b^2)*f*cosh(d*x + c)^2 - (a^2 + b^2)*f)*sinh(d*x + c)^2 + (a^2 + b^2)*f + 4*((a^2 + b^2)*f*cosh(d*x + c)^3 - (a^2 + b^2)*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2 )/b^2) - b)/b + 1) + ((a^2 + b^2)*f*cosh(d*x + c)^4 + 4*(a^2 + b^2)*f*cosh (d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)*f*sinh(d*x + c)^4 - 2*(a^2 + b^2)* f*cosh(d*x + c)^2 + 2*(3*(a^2 + b^2)*f*cosh(d*x + c)^2 - (a^2 + b^2)*f)*si nh(d*x + c)^2 + (a^2 + b^2)*f + 4*((a^2 + b^2)*f*cosh(d*x + c)^3 - (a^2 + b^2)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c...
\[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \coth \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-(a^2*d*integrate(x/(a^3*d*e^(d*x + c) + a^3*d), x) + b^2*d*integrate(x/(a ^3*d*e^(d*x + c) + a^3*d), x) - a^2*d*integrate(x/(a^3*d*e^(d*x + c) - a^3 *d), x) - b^2*d*integrate(x/(a^3*d*e^(d*x + c) - a^3*d), x) + a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1)/(a^3*d^2)) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) - (2*b*d*x*e^(3*d*x + 3*c) - 2*b*d*x*e^( d*x + c) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x) + a)/(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) - integrate(2*((a^3*e^c + a* b^2*e^c)*x*e^(d*x) - (a^2*b + b^3)*x)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d* x + c) - a^3*b), x))*f - e*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^( -3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - (a^ 2 + b^2)*log(e^(-d*x - c) + 1)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) - 1) /(a^3*d))
Timed out. \[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]