\(\int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [446]

Optimal result
Mathematica [A] (warning: unable to verify)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F]
Giac [F(-1)]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 32, antiderivative size = 746 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d} \] Output:

1/2*f*x/a/d-2*b^3*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d-b*(f*x+e)*arcta 
n(exp(d*x+c))/(a^2+b^2)/d-2*f*x*arctanh(exp(2*d*x+2*c))/a/d-b^4*(f*x+e)*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)*ln(1+b*ex 
p(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d+b^4*(f*x+e)*ln(1+exp(2*d*x+2 
*c))/a/(a^2+b^2)^2/d-f*x*ln(tanh(d*x+c))/a/d+(f*x+e)*ln(tanh(d*x+c))/a/d+1 
/2*I*b*f*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2-I*b^3*f*polylog(2,I*exp(d* 
x+c))/(a^2+b^2)^2/d^2-1/2*I*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+I*b^ 
3*f*polylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2-b^4*f*polylog(2,-b*exp(d*x+c) 
/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-b^4*f*polylog(2,-b*exp(d*x+c)/(a+( 
a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2+1/2*b^4*f*polylog(2,-exp(2*d*x+2*c))/a/ 
(a^2+b^2)^2/d^2-1/2*f*polylog(2,-exp(2*d*x+2*c))/a/d^2+1/2*f*polylog(2,exp 
(2*d*x+2*c))/a/d^2-1/2*b*f*sech(d*x+c)/(a^2+b^2)/d^2-1/2*b^2*(f*x+e)*sech( 
d*x+c)^2/a/(a^2+b^2)/d-1/2*f*tanh(d*x+c)/a/d^2+1/2*b^2*f*tanh(d*x+c)/a/(a^ 
2+b^2)/d^2-1/2*b*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d-1/2*(f*x+e)*t 
anh(d*x+c)^2/a/d
 

Mathematica [A] (warning: unable to verify)

Time = 10.44 (sec) , antiderivative size = 1080, normalized size of antiderivative = 1.45 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.77 (sec) , antiderivative size = 627, normalized size of antiderivative = 0.84, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {6123, 5985, 2009, 6107, 6107, 6095, 2620, 2715, 2838, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2620
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 5985
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + 
b*x]^p, x]}, Simp[(c + d*x)^m   u, x] - Simp[d*m   Int[(c + d*x)^(m - 1)*u, 
 x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n 
, p]
 

rule 6095
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6107
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2)   Int[(e + 
f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 
+ b^2)   Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 
]
 

rule 6123
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + 
 (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/a   Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ 
a   Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + 
d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] 
&& IGtQ[p, 0]
 

rule 7293
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 
Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2579 vs. \(2 (693 ) = 1386\).

Time = 24.97 (sec) , antiderivative size = 2580, normalized size of antiderivative = 3.46

method result size
risch \(\text {Expression too large to display}\) \(2580\)

Input:

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

Output:

4/d^2/(a^2+b^2)*c*a^2*f/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-8/d/(a^2+b^2)*b 
^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*x-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b 
^2)*ln(1+I*exp(d*x+c))*a*c-8/d/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d* 
x+c))*a*x-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*c+(-b*d 
*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-b*d*e*exp(3*d*x+3*c)+2*a*d*e* 
exp(2*d*x+2*c)+b*d*f*x*exp(d*x+c)-b*f*exp(3*d*x+3*c)+a*f*exp(2*d*x+2*c)+b* 
d*e*exp(d*x+c)-f*b*exp(d*x+c)+a*f)/d^2/(a^2+b^2)/(1+exp(2*d*x+2*c))^2-1/d/ 
(a^2+b^2)^2*b^4*e/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/2/d/(a^2+b^2)^ 
(3/2)*b^2*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2/(a^2+b 
^2)^2*b^4*f/a*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)) 
)-1/d^2/(a^2+b^2)^2*b^4*f/a*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2 
+b^2)^(1/2)))-12/d/(a^2+b^2)*b^3*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/2/d/ 
(a^2+b^2)^(5/2)*b^4*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/ 
d/(a^2+b^2)*a^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-4/d^2/(a^2+b^2)*a^3*f 
/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*d 
ilog(1-I*exp(d*x+c))-6*I/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d 
*x+c))+1/(a^2+b^2)/d*ln(exp(d*x+c)+1)*a*f*x-1/2/d^2/(a^2+b^2)^(5/2)*c*b^4* 
f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+12/d^2/(a^2+b^2)*c*b^3 
*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))-8/d/(a^2+b^2)*b^2*e/(4*a^2+4*b^2)*a*ln 
(1+exp(2*d*x+2*c))+1/2/d/(a^2+b^2)^(5/2)*b^2*e*a^2*arctanh(1/2*(2*b*exp...
 

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. 7645 vs. \(2 (676) = 1352\).

Time = 0.36 (sec) , antiderivative size = 7645, normalized size of antiderivative = 10.25 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + 
a*b^4)*d) - (a^2*b + 3*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)* 
d) + (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) 
 + (b*e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^ 
2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d) - lo 
g(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e - f*(((b*d*x*e^ 
(3*c) + b*e^(3*c))*e^(3*d*x) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x) - ( 
b*d*x*e^c - b*e^c)*e^(d*x) - a)/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^ 
2*d^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x) 
) - 16*integrate(-1/8*(a*b^4*x*e^(d*x + c) - b^5*x)/(a^5*b + 2*a^3*b^3 + a 
*b^5 - (a^5*b*e^(2*c) + 2*a^3*b^3*e^(2*c) + a*b^5*e^(2*c))*e^(2*d*x) - 2*( 
a^6*e^c + 2*a^4*b^2*e^c + a^2*b^4*e^c)*e^(d*x)), x) + 16*integrate(1/16*(( 
a^2*b*e^c + 3*b^3*e^c)*x*e^(d*x) - 2*(a^3 + 2*a*b^2)*x)/(a^4 + 2*a^2*b^2 + 
 b^4 + (a^4*e^(2*c) + 2*a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x) + 16 
*integrate(1/16*x/(a*e^(d*x + c) + a), x) - 16*integrate(1/16*x/(a*e^(d*x 
+ c) - a), x))
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((f*x+e)*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 - 3*e**(4*c + 4*d*x)*atan 
(e**(c + d*x))*a*b**3*e - 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*b*e - 
 6*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**3*e - atan(e**(c + d*x))*a**3* 
b*e - 3*atan(e**(c + d*x))*a*b**3*e + 32*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*f + 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**3*b**2*d*f + 32*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*b**4*d*f - e 
**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*e - 2*e**(4*c + 4*d*x)*log( 
e**(2*c + 2*d*x) + 1)*a**2*b**2*e + e**(4*c + 4*d*x)*log(e**(c + d*x) - 1) 
*a**4*e + 2*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**2*b**2*e + e**(4*c + 
 4*d*x)*log(e**(c + d*x) - 1)*b**4*e + e**(4*c + 4*d*x)*log(e**(c + d*x) + 
 1)*a**4*e + 2*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**2*b**2*e + e**(4* 
c + 4*d*x)*log(e**(c + d*x) + 1)*b**4*e - e**(4*c + 4*d*x)*log(e**(2*c ...