\(\int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [411]

Optimal result

Integrand size = 34, antiderivative size = 1294 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d-( 
f*x+e)^3*tanh(d*x+c)/b/d+6*I*a*f^3*polylog(3,I*exp(d*x+c))/b^2/d^4-6*a^3*f 
^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/ 
d^3+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+ 
b^2)^(3/2)/d^3+3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2 
)))/b/(a^2+b^2)^(3/2)/d^2-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^ 
2+b^2)^(1/2)))/b/(a^2+b^2)^(3/2)/d^2-a^4*(f*x+e)^3*tanh(d*x+c)/b^3/(a^2+b^ 
2)/d-a^3*(f*x+e)^3*sech(d*x+c)/b^2/(a^2+b^2)/d-a^4*(f*x+e)^3/b^3/(a^2+b^2) 
/d+a^2*(f*x+e)^3*tanh(d*x+c)/b^3/d+a*(f*x+e)^3*sech(d*x+c)/b^2/d+3*a^4*f^2 
*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^3/(a^2+b^2)/d^3+3*a^4*f*(f*x+e)^2*ln 
(1+exp(2*d*x+2*c))/b^3/(a^2+b^2)/d^2+6*a^3*f*(f*x+e)^2*arctan(exp(d*x+c))/ 
b^2/(a^2+b^2)/d^2-6*I*a^3*f^3*polylog(3,I*exp(d*x+c))/b^2/(a^2+b^2)/d^4-6* 
I*a*f^2*(f*x+e)*polylog(2,I*exp(d*x+c))/b^2/d^3+a^2*(f*x+e)^3/b^3/d-6*I*a^ 
3*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^3+3*f*(f*x+e)^2*ln( 
1+exp(2*d*x+2*c))/b/d^2+3*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b/d^3+6*I 
*a^3*f^3*polylog(3,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+6*I*a*f^2*(f*x+e)*poly 
log(2,-I*exp(d*x+c))/b^2/d^3-6*a*f*(f*x+e)^2*arctan(exp(d*x+c))/b^2/d^2-3/ 
2*a^4*f^3*polylog(3,-exp(2*d*x+2*c))/b^3/(a^2+b^2)/d^4-3*a^2*f^2*(f*x+e)*p 
olylog(2,-exp(2*d*x+2*c))/b^3/d^3-3*a^2*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b 
^3/d^2-6*I*a*f^3*polylog(3,-I*exp(d*x+c))/b^2/d^4+1/4*(f*x+e)^4/b/f-3/2...
 

Mathematica [A] (warning: unable to verify)

Time = 4.78 (sec) , antiderivative size = 1111, normalized size of antiderivative = 0.86 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(4*b) - (f*(12*b*d^3*e^2*E 
^(2*c)*x - 12*b*d^3*e^2*(1 + E^(2*c))*x - 12*b*d^3*e*f*x^2 - 4*b*d^3*f^2*x 
^3 + 12*a*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 6*b*d^2*e^2*(1 + E^( 
2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*a*d*e*(1 + E^(2*c))*f*(d 
*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^( 
c + d*x)] + PolyLog[2, I*E^(c + d*x)]) + 6*b*d*e*(1 + E^(2*c))*f*(2*d*x*(d 
*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]) + (6*I)*a*( 
1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c 
+ d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d 
*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3, I*E^(c + d*x)]) + b*( 
1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*P 
olyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))])))/(2*(a^2 + 
b^2)*d^4*(1 + E^(2*c))) + (a^3*(2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt 
[a^2 + b^2]] - 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]) 
] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^3*f 
^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e^2*f*x*Log[ 
1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^ 
(c + d*x))/(a + Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a 
 + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a 
+ Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 6115

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -(e+f x)^3 \tan (i c+i d x)^2dx}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\int (e+f x)^3 \tan (i c+i d x)^2dx}{b}\)

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 17

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \int i (e+f x)^2 \tanh (c+d x)dx}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {3 f \int (e+f x)^2 \tanh (c+d x)dx}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {-\frac {3 f \int -i (e+f x)^2 \tan (i c+i d x)dx}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \int (e+f x)^2 \tan (i c+i d x)dx}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 (c+d x)}\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 6101

\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 5974

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \int (e+f x)^2 \text {sech}(c+d x)dx}{d}-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (-\frac {2 i f \int (e+f x) \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 6117

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \tanh (c+d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \tanh (c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \tan (i c+i d x)dx}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 (c+d x)}\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (\frac {-\frac {(e+f x)^3 \text {sech}(c+d x)}{d}+\frac {3 f \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \tanh (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {(e+f x)^3 \tanh (c+d x)}{d}-\frac {(e+f x)^4}{4 f}}{b}\)

Input:

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

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

Input:

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

Output:

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

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. 7331 vs. \(2 (1195) = 2390\).

Time = 0.30 (sec) , antiderivative size = 7331, normalized size of antiderivative = 5.67 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral((e + f*x)**3*sinh(c + d*x)*tanh(c + d*x)**2/(a + b*sinh(c + d*x)) 
, x)
 

Maxima [F]

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

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

Output:

-3*b*e^2*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 
 + b^2)*d^2)) - 6*a*f^3*integrate(x^2*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + 
 b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 6*b*f^3*integrate(x^2/(a^2*d 
*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 12*a*e*f^2 
*integrate(x*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + 
a^2*d + b^2*d), x) - 12*b*e*f^2*integrate(x/(a^2*d*e^(2*d*x + 2*c) + b^2*d 
*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - (a^3*log((b*e^(-d*x - c) - a - sqr 
t(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^2*b + b^3)*sqrt( 
a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x 
 - 2*c))*d) - (d*x + c)/(b*d))*e^3 - 6*a*e^2*f*arctan(e^(d*x + c))/((a^2 + 
 b^2)*d^2) + 1/4*(24*b^2*e^2*f*x + (a^2*d*f^3 + b^2*d*f^3)*x^4 + 4*(a^2*d* 
e*f^2 + (d*e*f^2 + 2*f^3)*b^2)*x^3 + 6*(a^2*d*e^2*f + (d*e^2*f + 4*e*f^2)* 
b^2)*x^2 + ((a^2*d*f^3*e^(2*c) + b^2*d*f^3*e^(2*c))*x^4 + 4*(a^2*d*e*f^2*e 
^(2*c) + b^2*d*e*f^2*e^(2*c))*x^3 + 6*(a^2*d*e^2*f*e^(2*c) + b^2*d*e^2*f*e 
^(2*c))*x^2)*e^(2*d*x) + 8*(a*b*f^3*x^3*e^c + 3*a*b*e*f^2*x^2*e^c + 3*a*b* 
e^2*f*x*e^c)*e^(d*x))/(a^2*b*d + b^3*d + (a^2*b*d*e^(2*c) + b^3*d*e^(2*c)) 
*e^(2*d*x)) - integrate(-2*(a^3*f^3*x^3*e^c + 3*a^3*e*f^2*x^2*e^c + 3*a^3* 
e^2*f*x*e^c)*e^(d*x)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2 
*d*x) - 2*(a^3*b*e^c + a*b^3*e^c)*e^(d*x)), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**5*b*d**2*e**2*f - 48*e**(2*c 
 + 2*d*x)*atan(e**(c + d*x))*a**5*b*d*e*f**2 - 48*e**(2*c + 2*d*x)*atan(e* 
*(c + d*x))*a**5*b*f**3 - 48*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b**3 
*d**2*e**2*f - 96*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b**3*d*e*f**2 - 
 96*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b**3*f**3 - 24*e**(2*c + 2*d* 
x)*atan(e**(c + d*x))*a*b**5*d**2*e**2*f - 48*e**(2*c + 2*d*x)*atan(e**(c 
+ d*x))*a*b**5*d*e*f**2 - 48*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**5*f* 
*3 - 24*atan(e**(c + d*x))*a**5*b*d**2*e**2*f - 48*atan(e**(c + d*x))*a**5 
*b*d*e*f**2 - 48*atan(e**(c + d*x))*a**5*b*f**3 - 48*atan(e**(c + d*x))*a* 
*3*b**3*d**2*e**2*f - 96*atan(e**(c + d*x))*a**3*b**3*d*e*f**2 - 96*atan(e 
**(c + d*x))*a**3*b**3*f**3 - 24*atan(e**(c + d*x))*a*b**5*d**2*e**2*f - 4 
8*atan(e**(c + d*x))*a*b**5*d*e*f**2 - 48*atan(e**(c + d*x))*a*b**5*f**3 - 
 8*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*a**3*b**2*d**3*e**3*i - 8*sqrt(a**2 + b**2)*atan((e**(c + d*x 
)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**2*d**3*e**3*i - 32*e**(5*c + 2*d*x 
)*int((e**(3*d*x)*x**3)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4 
*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x) 
*a - b),x)*a**7*d**4*f**3 - 64*e**(5*c + 2*d*x)*int((e**(3*d*x)*x**3)/(e** 
(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a + e**(4*c + 4*d*x)*b + 4*e**(3*c + 
3*d*x)*a - e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**5*b**2*d**4...