\(\int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [478]

Optimal result

Integrand size = 32, antiderivative size = 298 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2} \] Output:

b*f*arctanh(cosh(d*x+c))/a^2/d^2-1/2*f*coth(d*x+c)/a/d^2+b*(f*x+e)*csch(d* 
x+c)/a^2/d-1/2*(f*x+e)*csch(d*x+c)^2/a/d-b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a- 
(a^2+b^2)^(1/2)))/a^3/d-b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2))) 
/a^3/d+b^2*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^3/d-b^2*f*polylog(2,-b*exp(d*x+c 
)/(a-(a^2+b^2)^(1/2)))/a^3/d^2-b^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^ 
(1/2)))/a^3/d^2+1/2*b^2*f*polylog(2,exp(2*d*x+2*c))/a^3/d^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(713\) vs. \(2(298)=596\).

Time = 8.22 (sec) , antiderivative size = 713, normalized size of antiderivative = 2.39 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {b \left (\frac {b (d e+d f x)^2}{2 f}+(b d e-a f+b d f x) \log \left (1-e^{-c-d x}\right )+(b d e+a f+b d f x) \log \left (1+e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right )}{a^3 d^2}-\frac {b^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a^3 d^2}+\frac {(d e-c f+f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )-a f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \] Input:

Integrate[((e + f*x)*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), 
x]
 

Output:

((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) + (b*((b*(d 
*e + d*f*x)^2)/(2*f) + (b*d*e - a*f + b*d*f*x)*Log[1 - E^(-c - d*x)] + (b* 
d*e + a*f + b*d*f*x)*Log[1 + E^(-c - d*x)] - b*f*PolyLog[2, -E^(-c - d*x)] 
 - b*f*PolyLog[2, E^(-c - d*x)]))/(a^3*d^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*Sinh[(c + d* 
x)/2] - 2*b*f*(c + d*x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.33 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.17, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6121, 5975, 3042, 25, 4254, 24, 6121, 5975, 3042, 26, 4257, 6103, 3042, 26, 4201, 2620, 2715, 2838, 6095, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6121

\(\displaystyle \frac {\int (e+f x) \coth (c+d x) \text {csch}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5975

\(\displaystyle \frac {\frac {f \int \text {csch}^2(c+d x)dx}{2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}+\frac {f \int -\csc (i c+i d x)^2dx}{2 d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}-\frac {f \int \csc (i c+i d x)^2dx}{2 d}}{a}\)

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}-\frac {i f \int 1d(-i \coth (c+d x))}{2 d^2}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 6121

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {\int (e+f x) \coth (c+d x) \text {csch}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 5975

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {\frac {f \int \text {csch}(c+d x)dx}{d}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}(c+d x)}{d}+\frac {f \int i \csc (i c+i d x)dx}{d}}{a}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}(c+d x)}{d}+\frac {i f \int \csc (i c+i d x)dx}{d}}{a}\right )}{a}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 6103

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \coth (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \coth (c+d x)}{2 d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 d}}{a}-\frac {b \left (\frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\right )}{a}\)

Input:

Int[((e + f*x)*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

(-1/2*(f*Coth[c + d*x])/d^2 - ((e + f*x)*Csch[c + d*x]^2)/(2*d))/a - (b*(( 
-((f*ArcTanh[Cosh[c + d*x]])/d^2) - ((e + f*x)*Csch[c + d*x])/d)/a - (b*(- 
((b*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt 
[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + 
b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/( 
b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)) 
)/a) - (I*(((-1/2*I)*(e + f*x)^2)/f + (2*I)*(((e + f*x)*Log[1 + E^(2*c - I 
*Pi + 2*d*x)])/(2*d) + (f*PolyLog[2, -E^(2*c - I*Pi + 2*d*x)])/(4*d^2))))/ 
a))/a))/a
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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]
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 5975

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]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6103

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]
 

rule 6121

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + 
 (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/a   Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Simp[b/ 
a   Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(a + b*Sinh[c + 
d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] 
&& IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(648\) vs. \(2(280)=560\).

Time = 1.33 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.18

method	result	size
risch	\(-\frac {-2 b d f x \,{\mathrm e}^{3 d x +3 c}+2 a d f x \,{\mathrm e}^{2 d x +2 c}-2 b d e \,{\mathrm e}^{3 d x +3 c}+2 a d e \,{\mathrm e}^{2 d x +2 c}+2 b d f x \,{\mathrm e}^{d x +c}+a f \,{\mathrm e}^{2 d x +2 c}+2 b d e \,{\mathrm e}^{d x +c}-a f}{d^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}+\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{3}}-\frac {b^{2} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{3}}-\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{2}}+\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{2}}-\frac {b^{2} c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{3}}+\frac {b^{2} c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a^{3}}-\frac {b^{2} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3}}-\frac {b^{2} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3}}+\frac {b^{2} f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d \,a^{3}}-\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3}}\)	\(649\)

Input:

int((f*x+e)*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVE 
RBOSE)
 

Output:

-(-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)/d^2/a^2/(exp(2*d*x+2*c)-1)^2-1/d^2*b^2/a^3*f*dilog(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(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d*b^2/a^3*e*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^2*b/a^2*f*ln(exp(d*x+c)-1)+1/d^2*b/a^2*f*ln(exp(d*x+c)+1)-1/ 
d^2*b^2/a^3*c*f*ln(exp(d*x+c)-1)+1/d^2*b^2/a^3*c*f*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*b^2/a^3*f*ln(exp(d*x+c)+1)*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*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
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2899 vs. \(2 (277) = 554\).

Time = 0.19 (sec) , antiderivative size = 2899, normalized size of antiderivative = 9.73 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="fricas")
 

Output:

(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) - (b^2*f*cosh(d*x + c 
)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^ 
2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x 
 + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*d 
ilog((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) - (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*c 
osh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + 
c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^ 
2*f*cosh(d*x + c)^3 - 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) + (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*si 
nh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 
2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + 
c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x 
+ c)) + (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b 
^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d 
*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*c...
 

Sympy [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \coth {\left (c + d x \right )} \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)**2/(a+b*sinh(d*x+c)),x)
 

Output:

Integral((e + f*x)*coth(c + d*x)*csch(c + d*x)**2/(a + b*sinh(c + d*x)), x 
)
 

Maxima [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="maxima")
 

Output:

-(4*b^2*d*integrate(1/4*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 4*b^2*d*integr 
ate(1/4*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) - 4*integrate(1/2*(a*b^2*x*e^(d*x + c) - 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) + b^2*log(-2*a*e^(-d*x - c) + b*e^(-2 
*d*x - 2*c) - b)/(a^3*d) - b^2*log(e^(-d*x - c) + 1)/(a^3*d) - b^2*log(e^( 
-d*x - c) - 1)/(a^3*d))
                                                                                    
                                                                                    
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm 
="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
 

Output:

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)*coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Output:

(96*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**5*d**2*f + 40*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**3*b**2*d**2*f - 64*e**(6*c + 4*d*x)*int((e**(2*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*d**2*f + 8*e**(5*c + 4*d*x) 
*int((e**(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**3*b**2*d**2*f + 
6*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*f - 2*e**(4*c + 4*d*x)*log(e 
**(c + d*x) - 1)*a**3*b*f + 3*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**4* 
d*e + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**4*f + 2*e**(4*c + 4*d*x) 
*log(e**(c + d*x) + 1)*a**3*b*f + 3*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1) 
*b**4*d*e - 3*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - 
 b)*b**4*d*e - 12*e**(4*c + 4*d*x)*a**4*d*f*x + 3*e**(4*c + 4*d*x)*a**4*f 
- 3*e**(4*c + 4*d*x)*a**2*b**2*d*e - 4*e**(3*c + 3*d*x)*a**3*b*f + 6*e*...