\(\int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [470]

Optimal result

Integrand size = 34, antiderivative size = 499 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b f \arctan (\sinh (c+d x))}{a^2 d^2}-\frac {b^3 f \arctan (\sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b f x \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {b f x \text {arctanh}(\cosh (c+d x))}{a^2 d}+\frac {b (e+f x) \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 (e+f x) \coth (2 c+2 d x)}{a d}+\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 f \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^2}+\frac {f \log (\sinh (2 c+2 d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b (e+f x) \text {sech}(c+d x)}{a^2 d}+\frac {b^3 (e+f x) \text {sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d} \] Output:

b*f*arctan(sinh(d*x+c))/a^2/d^2-b^3*f*arctan(sinh(d*x+c))/a^2/(a^2+b^2)/d^ 
2+2*b*f*x*arctanh(exp(d*x+c))/a^2/d-b*f*x*arctanh(cosh(d*x+c))/a^2/d+b*(f* 
x+e)*arctanh(cosh(d*x+c))/a^2/d-2*(f*x+e)*coth(2*d*x+2*c)/a/d+b^4*(f*x+e)* 
ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d-b^4*(f*x+e)*l 
n(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d-b^2*f*ln(cosh( 
d*x+c))/a/(a^2+b^2)/d^2+f*ln(sinh(2*d*x+2*c))/a/d^2+b*f*polylog(2,-exp(d*x 
+c))/a^2/d^2-b*f*polylog(2,exp(d*x+c))/a^2/d^2+b^4*f*polylog(2,-b*exp(d*x+ 
c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2-b^4*f*polylog(2,-b*exp(d*x 
+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2-b*(f*x+e)*sech(d*x+c)/a^2 
/d+b^3*(f*x+e)*sech(d*x+c)/a^2/(a^2+b^2)/d+b^2*(f*x+e)*tanh(d*x+c)/a/(a^2+ 
b^2)/d
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.25 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=4 \left (-\frac {i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{4 (a-i b) d^2}+\frac {i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{4 (a+i b) d^2}+\frac {\left (-d e \cosh \left (\frac {1}{2} (c+d x)\right )+c f \cosh \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {f \log (\cosh (c+d x))}{8 (a-i b) d^2}+\frac {f \log (\cosh (c+d x))}{8 (a+i b) d^2}+\frac {\frac {1}{2} b d^2 f x^2+(b d e+a f) (c+d x)-2 (b d e+a f-b c f) (c+d x)+2 b f (c+d x) \log \left (1+e^{-c-d x}\right )+2 (b d e+a f-b c f) \log \left (1+e^{c+d x}\right )-2 b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )}{8 a^2 d^2}+\frac {a f (c+d x)+2 (b d e-a f-b c f) (c+d x)-\frac {b (d e+d f x)^2}{2 f}-2 b f (c+d x) \log \left (1-e^{-c-d x}\right )-2 (b d e-a f-b c f) \log \left (1-e^{c+d x}\right )+2 b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{8 a^2 d^2}+\frac {b^4 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{4 a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-d e \sinh \left (\frac {1}{2} (c+d x)\right )+c f \sinh \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a d^2}+\frac {\text {sech}(c+d x) (-b d e+b c f-b f (c+d x)-a d e \sinh (c+d x)+a c f \sinh (c+d x)-a f (c+d x) \sinh (c+d x))}{4 \left (a^2+b^2\right ) d^2}\right ) \] Input:

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

Output:

4*(((-1/4*I)*f*ArcTan[Tanh[(c + d*x)/2]])/((a - I*b)*d^2) + ((I/4)*f*ArcTa 
n[Tanh[(c + d*x)/2]])/((a + I*b)*d^2) + ((-(d*e*Cosh[(c + d*x)/2]) + c*f*C 
osh[(c + d*x)/2] - f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(8*a* 
d^2) + (f*Log[Cosh[c + d*x]])/(8*(a - I*b)*d^2) + (f*Log[Cosh[c + d*x]])/( 
8*(a + I*b)*d^2) + ((b*d^2*f*x^2)/2 + (b*d*e + a*f)*(c + d*x) - 2*(b*d*e + 
 a*f - b*c*f)*(c + d*x) + 2*b*f*(c + d*x)*Log[1 + E^(-c - d*x)] + 2*(b*d*e 
 + a*f - b*c*f)*Log[1 + E^(c + d*x)] - 2*b*f*PolyLog[2, -E^(-c - d*x)])/(8 
*a^2*d^2) + (a*f*(c + d*x) + 2*(b*d*e - a*f - b*c*f)*(c + d*x) - (b*(d*e + 
 d*f*x)^2)/(2*f) - 2*b*f*(c + d*x)*Log[1 - E^(-c - d*x)] - 2*(b*d*e - a*f 
- b*c*f)*Log[1 - E^(c + d*x)] + 2*b*f*PolyLog[2, E^(-c - d*x)])/(8*a^2*d^2 
) + (b^4*(-2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*c*f*ArcT 
anh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*E^(c + d 
*x))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqr 
t[a^2 + b^2])] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - f* 
PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(4*a^2*(a^2 + b^2)^ 
(3/2)*d^2) + (Sech[(c + d*x)/2]*(-(d*e*Sinh[(c + d*x)/2]) + c*f*Sinh[(c + 
d*x)/2] - f*(c + d*x)*Sinh[(c + d*x)/2]))/(8*a*d^2) + (Sech[c + d*x]*(-(b* 
d*e) + b*c*f - b*f*(c + d*x) - a*d*e*Sinh[c + d*x] + a*c*f*Sinh[c + d*x] - 
 a*f*(c + d*x)*Sinh[c + d*x]))/(4*(a^2 + b^2)*d^2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.08 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.89, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {6123, 5984, 3042, 25, 4672, 26, 3042, 26, 3956, 6123, 5985, 2009, 6107, 3042, 3803, 25, 2694, 27, 2620, 2715, 2838, 7293, 2009}

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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

\(\displaystyle \frac {4 \int (e+f x) \text {csch}^2(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {4 \int -\left ((e+f x) \csc (2 i c+2 i d x)^2\right )dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \int (e+f x) \csc (2 i c+2 i d x)^2dx}{a}\)

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x) \coth (2 c+2 d x)}{2 d}-\frac {f \int \coth (2 c+2 d x)dx}{2 d}\right )}{a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-4*(((e + f*x)*Coth[2*c + 2*d*x])/(2*d) - (f*Log[(-I)*Sinh[2*c + 2*d*x]]) 
/(4*d^2)))/a - (b*((-(((e + f*x)*ArcTanh[Cosh[c + d*x]])/d) - f*(ArcTan[Si 
nh[c + d*x]]/d^2 + (2*x*ArcTanh[E^(c + d*x)])/d - (x*ArcTanh[Cosh[c + d*x] 
])/d + PolyLog[2, -E^(c + d*x)]/d^2 - PolyLog[2, E^(c + d*x)]/d^2) + ((e + 
 f*x)*Sech[c + d*x])/d)/a - (b*((-2*b^2*(-1/2*(b*(((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)))/Sqrt[a^2 + b^2] + (b*(((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)))/(2*Sqrt[a^2 + b^2])))/(a^2 + b 
^2) + (-((b*f*ArcTan[Sinh[c + d*x]])/d^2) - (a*f*Log[Cosh[c + d*x]])/d^2 + 
 (b*(e + f*x)*Sech[c + d*x])/d + (a*(e + f*x)*Tanh[c + d*x])/d)/(a^2 + b^2 
)))/a))/a
 

Defintions of rubi rules used

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 2694

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]
 

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 3803

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]
 

rule 3956

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

rule 4672

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]
 

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 5985

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + 
b*x]^p, x]}, Simp[(c + d*x)^m   u, x] - Simp[d*m   Int[(c + d*x)^(m - 1)*u, 
 x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n 
, p]
 

rule 6107

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2)   Int[(e + 
f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 
+ b^2)   Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 
]
 

rule 6123

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3716\) vs. \(2(476)=952\).

Time = 18.05 (sec) , antiderivative size = 3717, normalized size of antiderivative = 7.45

method	result	size
risch	\(\text {Expression too large to display}\)	\(3717\)

Input:

int((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURN 
VERBOSE)
 

Output:

1/2*(-2*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*(a^2+b^2)*a^2*b^ 
2*d*e-2*(a^2+b^2)^(3/2)*ln(exp(d*x+c)+1)*a^2*b*d*e-2*(a^2+b^2)^(3/2)*ln(ex 
p(d*x+c)-1)*a^2*b*c*f+2*(a^2+b^2)^(3/2)*ln(exp(d*x+c)-1)*a^2*b*d*e-14*exp( 
4*d*x+4*c)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^4*b^2*f-7*e 
xp(4*d*x+4*c)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^2*b^4*f+ 
2*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*ln(exp(d*x+c)+1)*a^3*f+2*exp(4*d*x+4*c)*( 
a^2+b^2)^(3/2)*ln(exp(d*x+c)-1)*a^3*f+2*dilog(exp(d*x+c))*exp(4*d*x+4*c)*( 
a^2+b^2)^(3/2)*b^3*f+2*dilog(exp(d*x+c)+1)*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)* 
b^3*f+2*ln(1+exp(2*d*x+2*c))*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*a^3*f-8*exp(4* 
d*x+4*c)*ln(exp(d*x+c))*(a^2+b^2)^(3/2)*a^3*f+8*exp(4*d*x+4*c)*arctanh(1/2 
*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*(a^2+b^2)*a^4*f+exp(4*d*x+4*c)*arct 
anh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*(a^2+b^2)*b^4*f-4*(a^2+b^2)^ 
(3/2)*a*b^2*d*e-8*(a^2+b^2)^(3/2)*a^3*d*f*x-2*(a^2+b^2)^(3/2)*ln(exp(d*x+c 
)+1)*a^2*b*d*f*x+2*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*ln(exp(d*x+c)-1)*b^3*c*f 
-2*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*ln(exp(d*x+c)-1)*b^3*d*e+2*dilog(exp(d*x 
+c))*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*a^2*b*f+2*dilog(exp(d*x+c)+1)*exp(4*d* 
x+4*c)*(a^2+b^2)^(3/2)*a^2*b*f+4*arctan(exp(d*x+c))*exp(4*d*x+4*c)*(a^2+b^ 
2)^(3/2)*a^2*b*f-4*exp(4*d*x+4*c)*ln(exp(d*x+c))*(a^2+b^2)^(3/2)*a*b^2*f+6 
*exp(4*d*x+4*c)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*(a^2+b^2 
)*a^2*b^2*f+2*exp(4*d*x+4*c)*(a^2+b^2)^(3/2)*ln(exp(d*x+c)+1)*a*b^2*f+2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4086 vs. \(2 (472) = 944\).

Time = 0.22 (sec) , antiderivative size = 4086, normalized size of antiderivative = 8.19 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(b^4*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt 
(a^2 + b^2)))/((a^4 + a^2*b^2)*sqrt(a^2 + b^2)*d) - 2*(a*b*e^(-d*x - c) + 
b^2*e^(-2*d*x - 2*c) - a*b*e^(-3*d*x - 3*c) + 2*a^2 + b^2)/((a^3 + a*b^2 - 
 (a^3 + a*b^2)*e^(-4*d*x - 4*c))*d) + b*log(e^(-d*x - c) + 1)/(a^2*d) - b* 
log(e^(-d*x - c) - 1)/(a^2*d))*e + (16*b^4*integrate(-1/8*x*e^(d*x + c)/(a 
^4*b + a^2*b^3 - (a^4*b*e^(2*c) + a^2*b^3*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c 
+ a^3*b^2*e^c)*e^(d*x)), x) - 16*b*d*integrate(1/16*x/(a^2*d*e^(d*x + c) + 
 a^2*d), x) - 16*b*d*integrate(1/16*x/(a^2*d*e^(d*x + c) - a^2*d), x) - a* 
((d*x + c)/(a^2*d^2) - log(e^(d*x + c) + 1)/(a^2*d^2)) - a*((d*x + c)/(a^2 
*d^2) - log(e^(d*x + c) - 1)/(a^2*d^2)) + 2*(a*b*x*e^(3*d*x + 3*c) + b^2*x 
*e^(2*d*x + 2*c) - a*b*x*e^(d*x + c) + (2*a^2 + b^2)*x)/(a^3*d + a*b^2*d - 
 (a^3*d*e^(4*c) + a*b^2*d*e^(4*c))*e^(4*d*x)) - 2*a*x/((a^2 + b^2)*d) + 2* 
b*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + a*log(e^(2*d*x + 2*c) + 1)/((a^2 
 + b^2)*d^2))*f
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*b**4*e*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/ 
sqrt(a**2 + b**2))*b**4*e*i + 32*e**(9*c + 4*d*x)*int((e**(5*d*x)*x)/(e**( 
10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 2*e**(6*c + 
 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2*c + 2*d*x) 
*b + 2*e**(c + d*x)*a - b),x)*a**6*d*f + 64*e**(9*c + 4*d*x)*int((e**(5*d* 
x)*x)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 
2*e**(6*c + 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2 
*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**2*d*f + 32*e**(9*c + 4*d* 
x)*int((e**(5*d*x)*x)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8 
*c + 8*d*x)*b - 2*e**(6*c + 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 
4*d*x)*b + e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**4*d*f - e 
**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*b*e - 2*e**(4*c + 4*d*x)*log(e* 
*(c + d*x) - 1)*a**2*b**3*e - e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**5* 
e + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**4*b*e + 2*e**(4*c + 4*d*x)*l 
og(e**(c + d*x) + 1)*a**2*b**3*e + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)* 
b**5*e - 4*e**(4*c + 4*d*x)*a**5*e - 6*e**(4*c + 4*d*x)*a**3*b**2*e - 2*e* 
*(4*c + 4*d*x)*a*b**4*e - 2*e**(3*c + 3*d*x)*a**4*b*e - 2*e**(3*c + 3*d*x) 
*a**2*b**3*e - 2*e**(2*c + 2*d*x)*a**3*b**2*e - 2*e**(2*c + 2*d*x)*a*b**4* 
e + 2*e**(c + d*x)*a**4*b*e + 2*e**(c + d*x)*a**2*b**3*e - 32*e**(5*c)*...