\(\int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [473]

Optimal result

Integrand size = 34, antiderivative size = 974 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

-b^5*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)^2/d-b*(f*x+e)*ln(tanh(d*x+ 
c))/a^2/d+b*f*x*ln(tanh(d*x+c))/a^2/d-3/2*(f*x+e)*arctan(sinh(d*x+c))/a/d+ 
3/2*f*x*arctan(sinh(d*x+c))/a/d+3/2*I*f*polylog(2,-I*exp(d*x+c))/a/d^2+1/2 
*I*b^2*f*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^2+I*b^4*f*polylog(2,I*exp(d 
*x+c))/a/(a^2+b^2)^2/d^2-f*arctanh(cosh(d*x+c))/a/d^2-(f*x+e)*csch(d*x+c)/ 
a/d-1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/a/d+b^5*f*polylog(2,-b*exp(d*x+c)/ 
(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^2/d^2+1/2*b*f*tanh(d*x+c)/a^2/d^2+1/2*b 
*(f*x+e)*tanh(d*x+c)^2/a^2/d-1/2*b*f*x/a^2/d-3/2*I*f*polylog(2,I*exp(d*x+c 
))/a/d^2+1/2*b^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/a/(a^2+b^2)/d-I*b^4*f*pol 
ylog(2,-I*exp(d*x+c))/a/(a^2+b^2)^2/d^2-1/2*I*b^2*f*polylog(2,-I*exp(d*x+c 
))/a/(a^2+b^2)/d^2-1/2*f*sech(d*x+c)/a/d^2-1/2*b*f*polylog(2,exp(2*d*x+2*c 
))/a^2/d^2-3*f*x*arctan(exp(d*x+c))/a/d+1/2*b*f*polylog(2,-exp(2*d*x+2*c)) 
/a^2/d^2+b^2*(f*x+e)*arctan(exp(d*x+c))/a/(a^2+b^2)/d-1/2*b^5*f*polylog(2, 
-exp(2*d*x+2*c))/a^2/(a^2+b^2)^2/d^2+2*b*f*x*arctanh(exp(2*d*x+2*c))/a^2/d 
+2*b^4*(f*x+e)*arctan(exp(d*x+c))/a/(a^2+b^2)^2/d-1/2*b^3*f*tanh(d*x+c)/a^ 
2/(a^2+b^2)/d^2+1/2*b^2*f*sech(d*x+c)/a/(a^2+b^2)/d^2+1/2*b^3*(f*x+e)*sech 
(d*x+c)^2/a^2/(a^2+b^2)/d+b^5*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2) 
))/a^2/(a^2+b^2)^2/d^2+b^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/ 
a^2/(a^2+b^2)^2/d+b^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/( 
a^2+b^2)^2/d
 

Mathematica [A] (warning: unable to verify)

Time = 11.17 (sec) , antiderivative size = 1437, normalized size of antiderivative = 1.48 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x] 
),x]
 

Output:

8*(((-(d*e*Cosh[(c + d*x)/2]) + c*f*Cosh[(c + d*x)/2] - f*(c + d*x)*Cosh[( 
c + d*x)/2])*Csch[(c + d*x)/2]*Csch[c + d*x]*(a + b*Sinh[c + d*x]))/(16*a* 
d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*(-1/2*(b*(d*e - c*f + f*(c + d 
*x))^2)/f + (-(b*d*e) + a*f + b*c*f - b*f*(c + d*x))*Log[1 - E^(-c - d*x)] 
 + (-(b*d*e) - a*f + b*c*f - b*f*(c + d*x))*Log[1 + E^(-c - d*x)] + b*f*Po 
lyLog[2, -E^(-c - d*x)] + b*f*PolyLog[2, E^(-c - d*x)])*(a + b*Sinh[c + d* 
x]))/(8*a^2*d^2*(b + a*Csch[c + d*x])) + (b^5*Csch[c + d*x]*(-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]))])*(a + b*Sinh 
[c + d*x]))/(16*a^2*(a^2 + b^2)^2*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d 
*x]*(-2*a^2*b*d*e*(c + d*x) - 4*b^3*d*e*(c + d*x) + 2*a^2*b*c*f*(c + d*x) 
+ 4*b^3*c*f*(c + d*x) - a^2*b*f*(c + d*x)^2 - 2*b^3*f*(c + d*x)^2 - 6*a^3* 
d*e*ArcTan[E^(c + d*x)] - 10*a*b^2*d*e*ArcTan[E^(c + d*x)] + 6*a^3*c*f*Arc 
Tan[E^(c + d*x)] + 10*a*b^2*c*f*ArcTan[E^(c + d*x)] - (3*I)*a^3*f*(c + ...
 

Rubi [A] (verified)

Time = 4.00 (sec) , antiderivative size = 803, normalized size of antiderivative = 0.82, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6123, 5985, 2009, 6123, 5985, 2009, 6107, 6107, 6095, 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.

\(\displaystyle \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6123

\(\displaystyle \frac {\int (e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle \frac {-f \int \left (\frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 \arctan (\sinh (c+d x))}{2 d}-\frac {3 \text {csch}(c+d x)}{2 d}\right )dx-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 6123

\(\displaystyle -\frac {b \left (\frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle -\frac {b \left (\frac {-f \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right )dx-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 6095

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}-f \left (\frac {2 \text {arctanh}\left (e^{2 c+2 d x}\right ) x}{d}+\frac {\log (\tanh (c+d x)) x}{d}-\frac {x}{2 d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {\left (\frac {\left (-\frac {(e+f x)^2}{2 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\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}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^2}{2 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2}}{a^2+b^2}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x) \text {sech}^2(c+d x)}{2 d}+\frac {a f \text {sech}(c+d x)}{2 d^2}+\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}-\frac {b f \tanh (c+d x)}{2 d^2}}{a^2+b^2}\right )}{a}\right )}{a}\)

Input:

Int[((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
 

Output:

((-3*(e + f*x)*ArcTan[Sinh[c + d*x]])/(2*d) - (3*(e + f*x)*Csch[c + d*x])/ 
(2*d) + ((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(2*d) - f*((3*x*ArcTan[E 
^(c + d*x)])/d - (3*x*ArcTan[Sinh[c + d*x]])/(2*d) + ArcTanh[Cosh[c + d*x] 
]/d^2 - (((3*I)/2)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (((3*I)/2)*PolyLog[ 
2, I*E^(c + d*x)])/d^2 + Sech[c + d*x]/(2*d^2)))/a - (b*((((e + f*x)*Log[T 
anh[c + d*x]])/d - ((e + f*x)*Tanh[c + d*x]^2)/(2*d) - f*(-1/2*x/d + (2*x* 
ArcTanh[E^(2*c + 2*d*x)])/d + (x*Log[Tanh[c + d*x]])/d + PolyLog[2, -E^(2* 
c + 2*d*x)]/(2*d^2) - PolyLog[2, E^(2*c + 2*d*x)]/(2*d^2) + Tanh[c + d*x]/ 
(2*d^2)))/a - (b*((b^2*((b^2*(-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 + Sqr 
t[a^2 + b^2]))])/(b*d^2)))/(a^2 + b^2) + ((b*(e + f*x)^2)/(2*f) + (2*a*(e 
+ f*x)*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)*Log[1 + E^(2*(c + d*x))])/d - 
 (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*a*f*PolyLog[2, I*E^(c + d*x 
)])/d^2 - (b*f*PolyLog[2, -E^(2*(c + d*x))])/(2*d^2))/(a^2 + b^2)))/(a^2 + 
 b^2) + ((a*(e + f*x)*ArcTan[E^(c + d*x)])/d - ((I/2)*a*f*PolyLog[2, (-I)* 
E^(c + d*x)])/d^2 + ((I/2)*a*f*PolyLog[2, I*E^(c + d*x)])/d^2 + (a*f*Sech[ 
c + d*x])/(2*d^2) + (b*(e + f*x)*Sech[c + d*x]^2)/(2*d) - (b*f*Tanh[c + d* 
x])/(2*d^2) + (a*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*d))/(a^2 + b...
 

Defintions of rubi rules used

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 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 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 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 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)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3279 vs. \(2 (902 ) = 1804\).

Time = 46.24 (sec) , antiderivative size = 3280, normalized size of antiderivative = 3.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(3280\)

Input:

int((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURN 
VERBOSE)
 

Output:

-10*I/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/(a^2+b 
^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-10*I/(a^2+b^2)/d*a*b^2* 
f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+10*I/(a^2+b^2)/d*a*b^2*f/(4*a^2+4*b^2 
)*ln(1+I*exp(d*x+c))*x+6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*dilog(1+I*exp 
(d*x+c))-6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+4/(a^ 
2+b^2)/d*a^2*b*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+4/(a^2+b^2)/d*a^2*b*f/ 
(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+4/(a^2+b^2)/d^2*a^2*b*f/(4*a^2+4*b^2)*l 
n(1+I*exp(d*x+c))*c+4/(a^2+b^2)/d^2*a^2*b*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c 
))*c+7/2/(a^2+b^2)^(5/2)/d^2*a*c*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a 
^2+b^2)^(1/2))-4/(a^2+b^2)/d^2*a^2*c*b*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c) 
)+6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-6*I/(a^2+b^2) 
/d^2*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/(a^2+b^2)/d^2*a*b^2*f/( 
4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-10*I/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2 
)*dilog(1-I*exp(d*x+c))+6*I/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x 
+c))*x-6*I/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+1/(a^2+b^2 
)^2/d^2/a^2*b^5*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2 
)))+1/(a^2+b^2)^2/d^2/a^2*b^5*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(- 
a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)/d/a^2*b^3*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d 
/a^2*b^3*e*ln(exp(d*x+c)-1)-12/(a^2+b^2)/d*a^3*e/(4*a^2+4*b^2)*arctan(exp( 
d*x+c))+1/(a^2+b^2)/d^2/a^2*b^3*f*dilog(exp(d*x+c))-1/(a^2+b^2)/d^2/a^2...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 15223 vs. \(2 (879) = 1758\).

Time = 0.52 (sec) , antiderivative size = 15223, normalized size of antiderivative = 15.63 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit 
hm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*csch(d*x+c)**2*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit 
hm="maxima")
 

Output:

(b^5*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^6 + 2*a^4*b^2 + a 
^2*b^4)*d) + (3*a^3 + 5*a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^ 
4)*d) + (a^2*b + 2*b^3)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4) 
*d) - (2*a*b*e^(-2*d*x - 2*c) - 2*a*b*e^(-4*d*x - 4*c) + (3*a^2 + 2*b^2)*e 
^(-d*x - c) + 2*(a^2 + 2*b^2)*e^(-3*d*x - 3*c) + (3*a^2 + 2*b^2)*e^(-5*d*x 
 - 5*c))/((a^3 + a*b^2 + (a^3 + a*b^2)*e^(-2*d*x - 2*c) - (a^3 + a*b^2)*e^ 
(-4*d*x - 4*c) - (a^3 + a*b^2)*e^(-6*d*x - 6*c))*d) - b*log(e^(-d*x - c) + 
 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e + (32*b*d*integrate(1/32* 
x/(a^2*d*e^(d*x + c) + a^2*d), x) - 32*b*d*integrate(1/32*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*d*x 
*e^(2*d*x + 2*c) - 2*(a^2*d*e^(3*c) + 2*b^2*d*e^(3*c))*x*e^(3*d*x) + a*b - 
 (a^2*e^(5*c) + (3*a^2*d*e^(5*c) + 2*b^2*d*e^(5*c))*x)*e^(5*d*x) - (2*a*b* 
d*x*e^(4*c) + a*b*e^(4*c))*e^(4*d*x) + (a^2*e^c - (3*a^2*d*e^c + 2*b^2*d*e 
^c)*x)*e^(d*x))/(a^3*d^2 + a*b^2*d^2 - (a^3*d^2*e^(6*c) + a*b^2*d^2*e^(6*c 
))*e^(6*d*x) - (a^3*d^2*e^(4*c) + a*b^2*d^2*e^(4*c))*e^(4*d*x) + (a^3*d^2* 
e^(2*c) + a*b^2*d^2*e^(2*c))*e^(2*d*x)) - 32*integrate(-1/16*(a*b^5*x*e^(d 
*x + c) - b^6*x)/(a^6*b + 2*a^4*b^3 + a^2*b^5 - (a^6*b*e^(2*c) + 2*a^4*b^3 
*e^(2*c) + a^2*b^5*e^(2*c))*e^(2*d*x) - 2*(a^7*e^c + 2*a^5*b^2*e^c + a^3*b 
^4*e^c)*e^(d*x)), x) - 32*integrate(1/32*((3*a^3*e^c + 5*a*b^2*e^c)*x*e...
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorit 
hm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((e + f*x)/(cosh(c + d*x)^3*sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
 

Output:

int((e + f*x)/(cosh(c + d*x)^3*sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)
 

Output:

( - 9*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a**5*b*d*e - 15*e**(6*c + 6*d*x) 
*atan(e**(c + d*x))*a**3*b**3*d*e - 9*e**(4*c + 4*d*x)*atan(e**(c + d*x))* 
a**5*b*d*e - 15*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b**3*d*e + 9*e**( 
2*c + 2*d*x)*atan(e**(c + d*x))*a**5*b*d*e + 15*e**(2*c + 2*d*x)*atan(e**( 
c + d*x))*a**3*b**3*d*e + 9*atan(e**(c + d*x))*a**5*b*d*e + 15*atan(e**(c 
+ d*x))*a**3*b**3*d*e - 384*e**(11*c + 6*d*x)*int((e**(5*d*x)*x)/(e**(12*c 
 + 12*d*x)*b + 2*e**(11*c + 11*d*x)*a + 2*e**(9*c + 9*d*x)*a - 3*e**(8*c + 
 8*d*x)*b - 4*e**(7*c + 7*d*x)*a - 4*e**(5*c + 5*d*x)*a + 3*e**(4*c + 4*d* 
x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**7*d**2*f - 768*e 
**(11*c + 6*d*x)*int((e**(5*d*x)*x)/(e**(12*c + 12*d*x)*b + 2*e**(11*c + 1 
1*d*x)*a + 2*e**(9*c + 9*d*x)*a - 3*e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x 
)*a - 4*e**(5*c + 5*d*x)*a + 3*e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 
 2*e**(c + d*x)*a - b),x)*a**5*b**2*d**2*f - 384*e**(11*c + 6*d*x)*int((e* 
*(5*d*x)*x)/(e**(12*c + 12*d*x)*b + 2*e**(11*c + 11*d*x)*a + 2*e**(9*c + 9 
*d*x)*a - 3*e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x)*a - 4*e**(5*c + 5*d*x) 
*a + 3*e**(4*c + 4*d*x)*b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x 
)*a**3*b**4*d**2*f + 256*e**(10*c + 6*d*x)*int((e**(4*d*x)*x)/(e**(12*c + 
12*d*x)*b + 2*e**(11*c + 11*d*x)*a + 2*e**(9*c + 9*d*x)*a - 3*e**(8*c + 8* 
d*x)*b - 4*e**(7*c + 7*d*x)*a - 4*e**(5*c + 5*d*x)*a + 3*e**(4*c + 4*d*x)* 
b + 2*e**(3*c + 3*d*x)*a + 2*e**(c + d*x)*a - b),x)*a**6*b*d**2*f + 512...