3.5.40 \(\int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [440]

3.5.40.1 Optimal result

Integrand size = 34, antiderivative size = 1164 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

output

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

3.5.40.2 Mathematica [A] (verified)

Time = 9.07 (sec) , antiderivative size = 1441, normalized size of antiderivative = 1.24 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]

input

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

output

4*(-1/8*(f*Csch[c + d*x]*(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))*Ar 
cTan[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))]) - P 
olyLog[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*PolyLog[2, -E^(2*(c + d*x))] + 3*Po 
lyLog[3, -E^(2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/((a^2 + b^2)*d^4*(1 + 
E^(2*c))*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*((e + f*x)^3*Log[1 - E^(c 
 + d*x)] - (e + f*x)^3*Log[1 + E^(c + d*x)] - (3*f*(d^2*(e + f*x)^2*PolyLo 
g[2, -E^(c + d*x)] - 2*d*f*(e + f*x)*PolyLog[3, -E^(c + d*x)] + 2*f^2*Poly 
Log[4, -E^(c + d*x)]))/d^3 + (3*f*(d^2*(e + f*x)^2*PolyLog[2, E^(c + d*x)] 
 - 2*d*f*(e + f*x)*PolyLog[3, E^(c + d*x)] + 2*f^2*PolyLog[4, E^(c + d*x)] 
))/d^3)*(a + b*Sinh[c + d*x]))/(4*a*d*(b + a*Csch[c + d*x])) - (b^3*Csch[c 
 + d*x]*(-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...
 

3.5.40.3 Rubi [A] (verified)

Time = 4.93 (sec) , antiderivative size = 991, normalized size of antiderivative = 0.85, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6123, 5985, 25, 6107, 3042, 3803, 25, 2694, 27, 2620, 3011, 7163, 2720, 7143, 7292, 27, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.5.40.3.1 Defintions of rubi rules used

rule 25

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

rule 27

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

rule 2009

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

rule 2620

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

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 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 3803

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

rule 5985

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

rule 6107

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

rule 6123

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

rule 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 7292

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

rule 7293

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

3.5.40.4 Maple [F]

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

input

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

output

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

3.5.40.5 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. 9707 vs. \(2 (1064) = 2128\).

Time = 0.51 (sec) , antiderivative size = 9707, normalized size of antiderivative = 8.34 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.5.40.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.5.40.7 Maxima [F]

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

input

integrate((f*x+e)^3*csch(d*x+c)*sech(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) - (b^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^3 + a*b^2)*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) + log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e 
^3 - 6*a*e^2*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - 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) + 2*(b*f^3*x^3 + 3*b*e*f^2*x^2 + 3* 
b*e^2*f*x + (a*f^3*x^3*e^c + 3*a*e*f^2*x^2*e^c + 3*a*e^2*f*x*e^c)*e^(d*x)) 
/(a^2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) - 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) + ...
 

3.5.40.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.5.40.9 Mupad [F(-1)]

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

input

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

output

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