Integrand size = 32, antiderivative size = 324 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \] Output:
1/2*b*(f*x+e)^2/a^2/f-1/2*(a^2+b^2)*(f*x+e)^2/a^2/b/f-f*arctanh(cosh(d*x+c ))/a/d^2-(f*x+e)*csch(d*x+c)/a/d+(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a ^2+b^2)^(1/2)))/a^2/b/d+(a^2+b^2)*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^( 1/2)))/a^2/b/d-b*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d+(a^2+b^2)*f*polylog(2, -b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^2+(a^2+b^2)*f*polylog(2,-b*exp( d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^2-1/2*b*f*polylog(2,exp(2*d*x+2*c))/a^ 2/d^2
Time = 6.68 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-\frac {b d^2 (e+f x)^2}{f}-a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )-2 (-a f+b d (e+f x)) \log \left (1-e^{-c-d x}\right )-2 (a f+b d (e+f x)) \log \left (1+e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+\frac {\left (a^2+b^2\right ) \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b}+a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \] Input:
Integrate[((e + f*x)*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x]
Output:
(-((b*d^2*(e + f*x)^2)/f) - a*d*(e + f*x)*Coth[(c + d*x)/2] - 2*(-(a*f) + b*d*(e + f*x))*Log[1 - E^(-c - d*x)] - 2*(a*f + b*d*(e + f*x))*Log[1 + E^( -c - d*x)] + 2*b*f*PolyLog[2, -E^(-c - d*x)] + 2*b*f*PolyLog[2, E^(-c - d* x)] + ((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]))]))/b + a*d*(e + f*x)*Tanh[(c + d*x)/2])/(2*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.
| \(\displaystyle \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {\int (e+f x) \cosh (c+d x) \coth ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {\int (e+f x) \cosh (c+d x)dx+\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx+\int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {i f \int -i \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \int \sinh (c+d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \int -i \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx+\frac {i f \int \sin (i c+i d x)dx}{d}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5975 |
| \(\displaystyle \frac {\frac {f \int \text {csch}(c+d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \int i \csc (i c+i d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {i f \int \csc (i c+i d x)dx}{d}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 6119 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \cosh ^2(c+d x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \coth (c+d x)dx+\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx+\int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {f \int \sinh ^2(c+d x)dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \int \sin (i c+i d x)^2dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 6099 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {f \cosh (c+d x)}{d^2}+\frac {(e+f x) \sinh (c+d x)}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\right )}{a}\) |
Input:
Int[((e + f*x)*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(937\) vs. \(2(306)=612\).
Time = 1.22 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.90
| method | result | size |
| risch | \(-\frac {f \,c^{2}}{d^{2} b}+\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d b}-\frac {2 e \ln \left ({\mathrm e}^{d x +c}\right )}{d b}-\frac {2 f c x}{d b}+\frac {2 c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {c b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {c b f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d^{2}}-\frac {f \,x^{2}}{2 b}-\frac {c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b}+\frac {f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b}+\frac {f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b}+\frac {f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d b}+\frac {f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d b}+\frac {e x}{b}+\frac {f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b}+\frac {f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b}-\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {b e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d}-\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}-\frac {2 \left (f x +e \right ) {\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(938\) |
Input:
int((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVE RBOSE)
Output:
-1/d^2/b*f*c^2+1/d/b*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d/b*e*ln(ex p(d*x+c))-2/d/b*f*c*x+2/d^2/b*c*f*ln(exp(d*x+c))+1/a/d^2*f*ln(exp(d*x+c)-1 )-1/a/d^2*f*ln(exp(d*x+c)+1)-1/a^2/d*b*f*ln(exp(d*x+c)+1)*x+1/a^2/d*b*f*ln ((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/a^2/d*b*f*ln( (b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/a^2/d^2*b*f*ln(( -b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/a^2/d^2*b*f*ln( (b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/a^2/d^2*c*b*f*ln (exp(d*x+c)-1)-1/a^2/d^2*c*b*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/2/b *f*x^2-1/d^2/b*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d^2/b*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*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*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*f*ln((b*exp(d*x+c)+(a^2+ b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/b*e*x+1/d^2/b*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*f*dilog((b*exp(d*x+c)+( a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/a^2/d^2*b*f*dilog(exp(d*x+c)+1)+1 /a^2/d^2*b*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2))) +1/a^2/d^2*b*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2))) +1/a^2/d^2*b*f*dilog(exp(d*x+c))-1/a^2/d*b*e*ln(exp(d*x+c)-1)+1/a^2/d*b*e* ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/a^2/d*b*e*ln(exp(d*x+c)+1)-2/d*(f* x+e)/a*exp(d*x+c)/(exp(2*d*x+2*c)-1)
Leaf count of result is larger than twice the leaf count of optimal. 1735 vs. \(2 (303) = 606\).
Time = 0.12 (sec) , antiderivative size = 1735, normalized size of antiderivative = 5.35 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="fricas")
Output:
1/2*(a^2*d^2*f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f - (a^2*d^2* f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f)*cosh(d*x + c)^2 - (a^2* d^2*f*x^2 + 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f)*sinh(d*x + c)^2 - 4 *(a*b*d*f*x + a*b*d*e)*cosh(d*x + c) + 2*((a^2 + b^2)*f*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f*sinh(d*x + c)^ 2 - (a^2 + b^2)*f)*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) + 2*((a^2 + b^2) *f*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f*sinh(d*x + c)^2 - (a^2 + b^2)*f)*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) - 2*(b^2*f*cosh(d*x + c)^2 + 2*b^2*f*cosh(d*x + c)*sinh(d*x + c) + b^ 2*f*sinh(d*x + c)^2 - b^2*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(b^2 *f*cosh(d*x + c)^2 + 2*b^2*f*cosh(d*x + c)*sinh(d*x + c) + b^2*f*sinh(d*x + c)^2 - b^2*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 2*((a^2 + b^2)*d*e - (a^2 + b^2)*c*f - ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2* b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*((a^2 + b^2)*d*e - (a^2 + b^2)*c*f - ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c)^2 - 2* ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2...
\[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh {\left (c + d x \right )} \coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)*cosh(d*x+c)*coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral((e + f*x)*cosh(c + d*x)*coth(c + d*x)**2/(a + b*sinh(c + d*x)), x )
\[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \coth \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="maxima")
Output:
1/2*(2*b*d*integrate(x/(a^2*d*e^(d*x + c) + a^2*d), x) - 2*b*d*integrate(x /(a^2*d*e^(d*x + c) - a^2*d), x) + 2*a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) + 1)/(a^2*d^2)) - 2*a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) - 1)/(a^2 *d^2)) + (a*d*x^2*e^(2*d*x + 2*c) - a*d*x^2 - 4*b*x*e^(d*x + c))/(a*b*d*e^ (2*d*x + 2*c) - a*b*d) - integrate(4*((a^3*e^c + a*b^2*e^c)*x*e^(d*x) - (a ^2*b + b^3)*x)/(a^2*b^2*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^2*b^2), x))*f + e*((d*x + c)/(b*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*b*d))
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {coth}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)),x)
Output:
int((cosh(c + d*x)*coth(c + d*x)^2*(e + f*x))/(a + b*sinh(c + d*x)), x)
\[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right ) \cosh \left (d x +c \right ) \coth \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x \] Input:
int((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
int((f*x+e)*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)