\(\int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [445]

Optimal result

Integrand size = 34, antiderivative size = 1176 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

-1/2*b^4*f^2*polylog(3,-exp(2*d*x+2*c))/a/(a^2+b^2)^2/d^3-1/2*b^2*(f*x+e)^ 
2*sech(d*x+c)^2/a/(a^2+b^2)/d-1/2*b*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/(a^2 
+b^2)/d-2*I*b^3*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)^2/d^3-I*b*f^2*polyl 
og(3,-I*exp(d*x+c))/(a^2+b^2)/d^3-2*I*b^3*f*(f*x+e)*polylog(2,I*exp(d*x+c) 
)/(a^2+b^2)^2/d^2-I*b*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+I*b* 
f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3+b^4*(f*x+e)^2*ln(1+exp(2*d*x+2*c 
))/a/(a^2+b^2)^2/d-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 
/2)))/a/(a^2+b^2)^2/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^ 
2)^(1/2)))/a/(a^2+b^2)^2/d^2-b^2*f^2*ln(cosh(d*x+c))/a/(a^2+b^2)/d^3-b*f*( 
f*x+e)*sech(d*x+c)/(a^2+b^2)/d^2+b*f^2*arctan(sinh(d*x+c))/(a^2+b^2)/d^3+f 
^2*ln(cosh(d*x+c))/a/d^3-1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3-2*(f*x+e) 
^2*arctanh(exp(2*d*x+2*c))/a/d+1/2*f^2*polylog(3,-exp(2*d*x+2*c))/a/d^3-f* 
(f*x+e)*tanh(d*x+c)/a/d^2+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^( 
1/2)))/a/(a^2+b^2)^2/d^3+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1 
/2)))/a/(a^2+b^2)^2/d^3-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2) 
))/a/(a^2+b^2)^2/d-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/ 
(a^2+b^2)^2/d-b*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d-2*b^3*(f*x+e)^2*a 
rctan(exp(d*x+c))/(a^2+b^2)^2/d+f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2+ 
2*I*b^3*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)^2/d^3+2*I*b^3*f*(f*x+e)*poly 
log(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2+I*b*f*(f*x+e)*polylog(2,-I*exp(d*x...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4072\) vs. \(2(1176)=2352\).

Time = 12.57 (sec) , antiderivative size = 4072, normalized size of antiderivative = 3.46 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/3*(E^(2*c)*((2*(e + f*x)^3)/(E^(2*c)*f) - (3*(1 - E^(-2*c))*(e + f*x)^2 
*Log[1 - E^(-c - d*x)])/d - (3*(1 - E^(-2*c))*(e + f*x)^2*Log[1 + E^(-c - 
d*x)])/d + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f* 
PolyLog[3, -E^(-c - d*x)]))/(d^3*E^(2*c)) + (6*(-1 + E^(2*c))*f*(d*(e + f* 
x)*PolyLog[2, E^(-c - d*x)] + f*PolyLog[3, E^(-c - d*x)]))/(d^3*E^(2*c)))) 
/(a*(-1 + E^(2*c))) - (-12*a^3*d^3*e^2*E^(2*c)*x - 24*a*b^2*d^3*e^2*E^(2*c 
)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2*x - 12*a^3*d^3*e*E^( 
2*c)*f*x^2 - 24*a*b^2*d^3*e*E^(2*c)*f*x^2 - 4*a^3*d^3*E^(2*c)*f^2*x^3 - 8* 
a*b^2*d^3*E^(2*c)*f^2*x^3 + 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 18*b^3*d 
^2*e^2*ArcTan[E^(c + d*x)] + 6*a^2*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 
 18*b^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^(c + d 
*x)] - 12*b^3*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTan[E^(c + 
 d*x)] - 12*b^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] + (6*I)*a^2*b*d^2*e*f*x*Lo 
g[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I) 
*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2*e*E^(2*c) 
*f*x*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x 
)] + (9*I)*b^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c 
)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 - I 
*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*b^3* 
d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 ...
 

Rubi [A] (verified)

Time = 4.77 (sec) , antiderivative size = 993, normalized size of antiderivative = 0.84, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6123, 5985, 27, 6107, 6107, 6095, 2620, 3011, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(((e + f*x)^2*Log[Tanh[c + d*x]])/d - ((e + f*x)^2*Tanh[c + d*x]^2)/(2*d) 
- (f*(-1/2*(e + f*x)^2/f + (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/f - (f 
*Log[Cosh[c + d*x]])/d^2 + ((e + f*x)^2*Log[Tanh[c + d*x]])/f + ((e + f*x) 
*PolyLog[2, -E^(2*c + 2*d*x)])/d - ((e + f*x)*PolyLog[2, E^(2*c + 2*d*x)]) 
/d - (f*PolyLog[3, -E^(2*c + 2*d*x)])/(2*d^2) + (f*PolyLog[3, E^(2*c + 2*d 
*x)])/(2*d^2) + ((e + f*x)*Tanh[c + d*x])/d))/d)/a - (b*((b^2*((b^2*(-1/3* 
(e + f*x)^3/(b*f) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b 
^2])])/(b*d) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] 
)/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + 
b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d^2 
))/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + 
 b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d^ 
2))/(b*d)))/(a^2 + b^2) + ((b*(e + f*x)^3)/(3*f) + (2*a*(e + f*x)^2*ArcTan 
[E^(c + d*x)])/d - (b*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/d - ((2*I)*a*f 
*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + ((2*I)*a*f*(e + f*x)*PolyLo 
g[2, I*E^(c + d*x)])/d^2 - (b*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/d^ 
2 + ((2*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/d^3 - ((2*I)*a*f^2*PolyLog[ 
3, I*E^(c + d*x)])/d^3 + (b*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*d^3))/(a^ 
2 + b^2)))/(a^2 + b^2) + ((a*(e + f*x)^2*ArcTan[E^(c + d*x)])/d - (a*f^2*A 
rcTan[Sinh[c + d*x]])/d^3 + (b*f^2*Log[Cosh[c + d*x]])/d^3 - (I*a*f*(e ...
 

Defintions of rubi rules used

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 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 5985

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

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6107

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

rule 6123

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

rule 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 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] 
]
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(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. 16670 vs. \(2 (1088) = 2176\).

Time = 0.54 (sec) , antiderivative size = 16670, normalized size of antiderivative = 14.18 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-a^2*b*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2* 
b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^ 
2 + b^4*d^2), x) - 3*b^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d 
*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4* 
d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 2*a^3*d^2*f^2*integrate(x^2/(a^4*d^2* 
e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) 
+ a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a*b^2*d^2*f^2*integrate(x^2/( 
a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x 
 + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 2*a^2*b*d^2*e*f*integra 
te(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) 
+ b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 6*b^3 
*d^2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2* 
e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4* 
d^2), x) + 4*a^3*d^2*e*f*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2* 
d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + 
b^4*d^2), x) + 8*a*b^2*d^2*e*f*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^ 
2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2* 
d^2 + b^4*d^2), x) - a^3*f^2*(2*(d*x + c)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - 
log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - a*b^2*f^2*(2*(d* 
x + c)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 +...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b*e**2 - 3*e**(4*c + 4*d*x)*a 
tan(e**(c + d*x))*a*b**3*e**2 - 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3 
*b*e**2 - 6*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**3*e**2 - atan(e**(c + 
 d*x))*a**3*b*e**2 - 3*atan(e**(c + d*x))*a*b**3*e**2 + 32*e**(9*c + 4*d*x 
)*int((e**(5*d*x)*x**2)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a + e** 
(8*c + 8*d*x)*b + 4*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*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*d*f**2 + 64*e**(9*c + 4*d*x)*int((e**(5*d*x)*x**2)/(e**(10*c 
+ 10*d*x)*b + 2*e**(9*c + 9*d*x)*a + e**(8*c + 8*d*x)*b + 4*e**(7*c + 7*d* 
x)*a - 2*e**(6*c + 6*d*x)*b - 2*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**3*b**2*d*f**2 + 32*e**( 
9*c + 4*d*x)*int((e**(5*d*x)*x**2)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d 
*x)*a + e**(8*c + 8*d*x)*b + 4*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 
 2*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*b**4*d*f**2 + 64*e**(9*c + 4*d*x)*int((e**(5*d*x)*x)/ 
(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a + e**(8*c + 8*d*x)*b + 4*e**( 
7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*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*d*e*f + 128 
*e**(9*c + 4*d*x)*int((e**(5*d*x)*x)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9 
*d*x)*a + e**(8*c + 8*d*x)*b + 4*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x...