\(\int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [435]

Optimal result

Integrand size = 32, antiderivative size = 1049 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

b^2*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)/d+6*b^2*f^2*(f*x+e)*polylog 
(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^3+6*b^2*f^2*(f*x+e)*po 
lylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^3-3*b^2*f*(f*x+e) 
^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2-3*b^2*f*(f 
*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2+3/2*f 
*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2-3/2*f^2*(f*x+e)*polylog(3,exp(2 
*d*x+2*c))/a/d^3+3/2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a/d^3-3/2*f*(f 
*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/d^2-2*b*(f*x+e)^3*arctan(exp(d*x+c))/ 
(a^2+b^2)/d+3/4*f^3*polylog(4,exp(2*d*x+2*c))/a/d^4-3/4*f^3*polylog(4,-exp 
(2*d*x+2*c))/a/d^4-2*(f*x+e)^3*arctanh(exp(2*d*x+2*c))/a/d-b^2*(f*x+e)^3*l 
n(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-b^2*(f*x+e)^3*ln(1+b*e 
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-6*b^2*f^3*polylog(4,-b*exp(d* 
x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^4-6*b^2*f^3*polylog(4,-b*exp(d*x+c 
)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^4+6*I*b*f^2*(f*x+e)*polylog(3,I*exp(d 
*x+c))/(a^2+b^2)/d^3+6*I*b*f^3*polylog(4,-I*exp(d*x+c))/(a^2+b^2)/d^4+3*I* 
b*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2+3/4*b^2*f^3*polylog(4 
,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^4-6*I*b*f^3*polylog(4,I*exp(d*x+c))/(a^2+b 
^2)/d^4-3/2*b^2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^3+3/2 
*b^2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^2-6*I*b*f^2*(f*x 
+e)*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3-3*I*b*f*(f*x+e)^2*polylog(2,...
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3572\) vs. \(2(1049)=2098\).

Time = 12.45 (sec) , antiderivative size = 3572, normalized size of antiderivative = 3.41 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

2*((E^c*((e + f*x)^4/(4*E^c*f) + ((1 + E^(-c))*(e + f*x)^3*Log[1 + E^(-c - 
 d*x)])/d - (3*(1 + E^c)*f*(d^2*(e + f*x)^2*PolyLog[2, -E^(-c - d*x)] + 2* 
f*(d*(e + f*x)*PolyLog[3, -E^(-c - d*x)] + f*PolyLog[4, -E^(-c - d*x)])))/ 
(d^4*E^c)))/(2*a*(1 + E^c)) - ((e + f*x)^4/f - (4*(-1 + E^c)*(e + f*x)^3*L 
og[1 - E^(-c - d*x)])/d + (12*(-1 + E^c)*f*(d^2*(e + f*x)^2*PolyLog[2, E^( 
-c - d*x)] + 2*f*(d*(e + f*x)*PolyLog[3, E^(-c - d*x)] + f*PolyLog[4, E^(- 
c - d*x)])))/d^4)/(8*a*(-1 + E^c)) - (-8*a*d^4*e^3*E^(2*c)*x - 12*a*d^4*e^ 
2*E^(2*c)*f*x^2 - 8*a*d^4*e*E^(2*c)*f^2*x^3 - 2*a*d^4*E^(2*c)*f^3*x^4 + 8* 
b*d^3*e^3*ArcTan[E^(c + d*x)] + 8*b*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] + 
(12*I)*b*d^3*e^2*f*x*Log[1 - I*E^(c + d*x)] + (12*I)*b*d^3*e^2*E^(2*c)*f*x 
*Log[1 - I*E^(c + d*x)] + (12*I)*b*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)] + 
(12*I)*b*d^3*e*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (4*I)*b*d^3*f^3*x^ 
3*Log[1 - I*E^(c + d*x)] + (4*I)*b*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d* 
x)] - (12*I)*b*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*b*d^3*e^2*E^(2* 
c)*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*b*d^3*e*f^2*x^2*Log[1 + I*E^(c + d* 
x)] - (12*I)*b*d^3*e*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (4*I)*b*d^3* 
f^3*x^3*Log[1 + I*E^(c + d*x)] - (4*I)*b*d^3*E^(2*c)*f^3*x^3*Log[1 + I*E^( 
c + d*x)] + 4*a*d^3*e^3*Log[1 + E^(2*(c + d*x))] + 4*a*d^3*e^3*E^(2*c)*Log 
[1 + E^(2*(c + d*x))] + 12*a*d^3*e^2*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*d 
^3*e^2*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*d^3*e*f^2*x^2*Log[1 ...
 

Rubi [A] (verified)

Time = 4.58 (sec) , antiderivative size = 910, normalized size of antiderivative = 0.87, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {6123, 5984, 3042, 26, 4670, 3011, 6107, 6095, 2620, 3011, 7163, 2720, 7143, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 i \left (\frac {i \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^3}{d}-\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}+\frac {3 i f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{2 d}\right )}{a}-\frac {b \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^4}{4 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{d}-\frac {3 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{d^2}+\frac {3 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{d^2}-\frac {3 b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 d^2}+\frac {6 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{d^3}-\frac {6 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{d^3}+\frac {3 b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^4}}{a^2+b^2}\right )}{a}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[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 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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

rule 7293

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

Maple [F]

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

Input:

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

Output:

int((f*x+e)^3*csch(d*x+c)*sech(d*x+c)/(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. 2448 vs. \(2 (962) = 1924\).

Time = 0.18 (sec) , antiderivative size = 2448, normalized size of antiderivative = 2.33 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-(6*b^2*f^3*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + 
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*b^2*f^3*polylog(4, (a* 
cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt 
((a^2 + b^2)/b^2))/b) - 6*(a^2 + b^2)*f^3*polylog(4, cosh(d*x + c) + sinh( 
d*x + c)) - 6*(a^2 + b^2)*f^3*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) + 
 3*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2*f)*dilog((a*cosh(d*x 
 + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + 
b^2)/b^2) - b)/b + 1) + 3*(b^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e 
^2*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh 
(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 3*((a^2 + b^2)*d^2*f^3*x^2 
+ 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*dilog(cosh(d*x + c) + 
 sinh(d*x + c)) + 3*(a^2*d^2*f^3*x^2 + I*a*b*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2 
*x + 2*I*a*b*d^2*e*f^2*x + a^2*d^2*e^2*f + I*a*b*d^2*e^2*f)*dilog(I*cosh(d 
*x + c) + I*sinh(d*x + c)) + 3*(a^2*d^2*f^3*x^2 - I*a*b*d^2*f^3*x^2 + 2*a^ 
2*d^2*e*f^2*x - 2*I*a*b*d^2*e*f^2*x + a^2*d^2*e^2*f - I*a*b*d^2*e^2*f)*dil 
og(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 3*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a 
^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*dilog(-cosh(d*x + c) - sinh 
(d*x + c)) + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^ 
3*f^3)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^ 
2) + 2*a) + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-e^3*(b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d 
) - 2*b*arctan(e^(-d*x - c))/((a^2 + b^2)*d) + a*log(e^(-2*d*x - 2*c) + 1) 
/((a^2 + b^2)*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a* 
d)) + 3*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e^2*f/(a*d^2) + 3 
*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*e^2*f/(a*d^2) + 3*(d^2*x 
^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x 
 + c)))*e*f^2/(a*d^3) + 3*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^( 
d*x + c)) - 2*polylog(3, e^(d*x + c)))*e*f^2/(a*d^3) + (d^3*x^3*log(e^(d*x 
 + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c) 
) + 6*polylog(4, -e^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 
1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*poly 
log(4, e^(d*x + c)))*f^3/(a*d^4) - 1/2*(d^4*f^3*x^4 + 4*d^4*e*f^2*x^3 + 6* 
d^4*e^2*f*x^2)/(a*d^4) + integrate(2*(b^3*f^3*x^3 + 3*b^3*e*f^2*x^2 + 3*b^ 
3*e^2*f*x - (a*b^2*f^3*x^3*e^c + 3*a*b^2*e*f^2*x^2*e^c + 3*a*b^2*e^2*f*x*e 
^c)*e^(d*x))/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 
2*(a^4*e^c + a^2*b^2*e^c)*e^(d*x)), x) - integrate(-2*(a*f^3*x^3 + 3*a*e*f 
^2*x^2 + 3*a*e^2*f*x - (b*f^3*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f*x*e^ 
c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 \mathit {atan} \left (e^{d x +c}\right ) a b \,e^{3}+8 e^{3 c} \left (\int \frac {e^{3 d x} x^{3}}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a^{3} d \,f^{3}+8 e^{3 c} \left (\int \frac {e^{3 d x} x^{3}}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a \,b^{2} d \,f^{3}+24 e^{3 c} \left (\int \frac {e^{3 d x} x^{2}}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a^{3} d e \,f^{2}+24 e^{3 c} \left (\int \frac {e^{3 d x} x^{2}}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a \,b^{2} d e \,f^{2}+24 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a^{3} d \,e^{2} f +24 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -e^{4 d x +4 c} b -e^{2 d x +2 c} b -2 e^{d x +c} a +b}d x \right ) a \,b^{2} d \,e^{2} f -\mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{2} e^{3}+\mathrm {log}\left (e^{d x +c}-1\right ) a^{2} e^{3}+\mathrm {log}\left (e^{d x +c}-1\right ) b^{2} e^{3}+\mathrm {log}\left (e^{d x +c}+1\right ) a^{2} e^{3}+\mathrm {log}\left (e^{d x +c}+1\right ) b^{2} e^{3}-\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2} e^{3}}{a d \left (a^{2}+b^{2}\right )} \] Input:

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

Output:

( - 2*atan(e**(c + d*x))*a*b*e**3 + 8*e**(3*c)*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 - e**(2*c + 2*d 
*x)*b - 2*e**(c + d*x)*a + b),x)*a**3*d*f**3 + 8*e**(3*c)*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 - e* 
*(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a*b**2*d*f**3 + 24*e**(3*c)*in 
t((e**(3*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - e**(4*c + 
 4*d*x)*b - e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**3*d*e*f**2 + 
24*e**(3*c)*int((e**(3*d*x)*x**2)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x) 
*a - e**(4*c + 4*d*x)*b - e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a* 
b**2*d*e*f**2 + 24*e**(3*c)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e** 
(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b - e**(2*c + 2*d*x)*b - 2*e**(c + d*x) 
*a + b),x)*a**3*d*e**2*f + 24*e**(3*c)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x 
)*b + 2*e**(5*c + 5*d*x)*a - e**(4*c + 4*d*x)*b - e**(2*c + 2*d*x)*b - 2*e 
**(c + d*x)*a + b),x)*a*b**2*d*e**2*f - log(e**(2*c + 2*d*x) + 1)*a**2*e** 
3 + log(e**(c + d*x) - 1)*a**2*e**3 + log(e**(c + d*x) - 1)*b**2*e**3 + lo 
g(e**(c + d*x) + 1)*a**2*e**3 + log(e**(c + d*x) + 1)*b**2*e**3 - log(e**( 
2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*b**2*e**3)/(a*d*(a**2 + b**2))