\(\int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [437]

Optimal result

Integrand size = 30, antiderivative size = 439 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac {b^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2} \] Output:

-2*b*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)/d-2*(f*x+e)*arctanh(exp(2*d*x+2* 
c))/a/d-b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d-b 
^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d+b^2*(f*x+e 
)*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)/d+I*b*f*polylog(2,-I*exp(d*x+c))/(a^2+b 
^2)/d^2-I*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2-b^2*f*polylog(2,-b*exp 
(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2-b^2*f*polylog(2,-b*exp(d*x+c) 
/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)/d^2+1/2*b^2*f*polylog(2,-exp(2*d*x+2*c)) 
/a/(a^2+b^2)/d^2-1/2*f*polylog(2,-exp(2*d*x+2*c))/a/d^2+1/2*f*polylog(2,ex 
p(2*d*x+2*c))/a/d^2
 

Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.79 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {\frac {1}{2} d^2 f x^2+d e (c+d x)-2 (d e-c f) (c+d x)+2 f (c+d x) \log \left (1+e^{-c-d x}\right )+2 (d e-c f) \log \left (1+e^{c+d x}\right )-2 f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )}{a}+\frac {\frac {1}{2} d^2 f x^2-d e (c+d x)+2 c f (c+d x)+2 f (c+d x) \log \left (1-e^{-c-d x}\right )+2 (d e-c f) \log \left (1-e^{c+d x}\right )-2 f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a}-\frac {b^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a \left (a^2+b^2\right )}-\frac {2 \left (-a d e (c+d x)+a c f (c+d x)-\frac {1}{2} a f (c+d x)^2+2 b d e \arctan \left (e^{c+d x}\right )-2 b c f \arctan \left (e^{c+d x}\right )+i b f (c+d x) \log \left (1-i e^{c+d x}\right )-i b f (c+d x) \log \left (1+i e^{c+d x}\right )+a d e \log \left (1+e^{2 (c+d x)}\right )-a c f \log \left (1+e^{2 (c+d x)}\right )+a f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {1}{2} a f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )}{a^2+b^2}}{2 d^2} \] Input:

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

Output:

(((d^2*f*x^2)/2 + d*e*(c + d*x) - 2*(d*e - c*f)*(c + d*x) + 2*f*(c + d*x)* 
Log[1 + E^(-c - d*x)] + 2*(d*e - c*f)*Log[1 + E^(c + d*x)] - 2*f*PolyLog[2 
, -E^(-c - d*x)])/a + ((d^2*f*x^2)/2 - d*e*(c + d*x) + 2*c*f*(c + d*x) + 2 
*f*(c + d*x)*Log[1 - E^(-c - d*x)] + 2*(d*e - c*f)*Log[1 - E^(c + d*x)] - 
2*f*PolyLog[2, E^(-c - d*x)])/a - (b^2*(-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))/Sqr 
t[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTa 
nh[(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]))]))/(a*(a^2 + b^2)) - (2*(-(a*d*e*( 
c + d*x)) + a*c*f*(c + d*x) - (a*f*(c + d*x)^2)/2 + 2*b*d*e*ArcTan[E^(c + 
d*x)] - 2*b*c*f*ArcTan[E^(c + d*x)] + I*b*f*(c + d*x)*Log[1 - I*E^(c + d*x 
)] - I*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + a*d*e*Log[1 + E^(2*(c + d*x) 
)] - a*c*f*Log[1 + E^(2*(c + d*x))] + a*f*(c + d*x)*Log[1 + E^(2*(c + d*x) 
)] - I*b*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*f*PolyLog[2, I*E^(c + d*x)] 
+ (a*f*PolyLog[2, -E^(2*(c + d*x))])/2))/(a^2 + b^2))/(2*d^2)
 

Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.92, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6123, 5984, 3042, 26, 4670, 2715, 2838, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((b*((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*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*Pol 
yLog[2, -E^(2*(c + d*x))])/(2*d^2))/(a^2 + b^2)))/a) + ((2*I)*((I*(e + f*x 
)*ArcTanh[E^(2*c + 2*d*x)])/d + ((I/4)*f*PolyLog[2, -E^(2*c + 2*d*x)])/d^2 
 - ((I/4)*f*PolyLog[2, E^(2*c + 2*d*x)])/d^2))/a
 

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 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 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 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. 1064 vs. \(2 (411 ) = 822\).

Time = 4.75 (sec) , antiderivative size = 1065, normalized size of antiderivative = 2.43

method	result	size
risch	\(\text {Expression too large to display}\)	\(1065\)

Input:

int((f*x+e)*csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERB 
OSE)
 

Output:

1/d^2*c*f*b^2/(a^2+b^2)/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d*f*b^2/ 
(a^2+b^2)/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1 
/d^2*f*b^2/(a^2+b^2)/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^ 
(1/2)))*c-1/d*f*b^2/(a^2+b^2)/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^ 
2+b^2)^(1/2)))*x-1/d^2*f*b^2/(a^2+b^2)/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+ 
a)/(a+(a^2+b^2)^(1/2)))*c-4*I/d^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c+4 
*I/d^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c-4*I/d*f/(4*a^2+4*b^2)*ln(1-I 
*exp(d*x+c))*b*x+4*I/d*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x-1/d^2*c*f/a* 
ln(exp(d*x+c)-1)+1/d*f/a*ln(exp(d*x+c)+1)*x-1/d^2*f*b^2/(a^2+b^2)/a*dilog( 
(-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*f*b^2/(a^2+b 
^2)/a*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-4/d*f/(4 
*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*x-4/d^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c) 
)*a*c-4/d*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*x-4/d^2*f/(4*a^2+4*b^2)*ln( 
1-I*exp(d*x+c))*a*c+1/d*e/a*ln(exp(d*x+c)-1)+1/d*e/a*ln(exp(d*x+c)+1)+1/d^ 
2*f/a*dilog(exp(d*x+c)+1)-1/d^2*f*dilog(exp(d*x+c))/a-4/d*e/(4*a^2+4*b^2)* 
a*ln(1+exp(2*d*x+2*c))-8/d*e/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-4/d^2*f/(4 
*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*a-4/d^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d* 
x+c))*a+4/d^2*c*f/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))+8/d^2*c*f/(4*a^2+4* 
b^2)*b*arctan(exp(d*x+c))-1/d*e*b^2/(a^2+b^2)/a*ln(b*exp(2*d*x+2*c)+2*a*ex 
p(d*x+c)-b)-4*I/d^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b+4*I/d^2*f/(...
 

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. 808 vs. \(2 (400) = 800\).

Time = 0.13 (sec) , antiderivative size = 808, normalized size of antiderivative = 1.84 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-(b^2*f*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*si 
nh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + b^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) - (a^2 + b^2)*f*dilog(cosh(d*x + c) + sinh(d*x + c)) - 
 (a^2 + b^2)*f*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (a^2*f + I*a*b*f)*d 
ilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (a^2*f - I*a*b*f)*dilog(-I*cosh( 
d*x + c) - I*sinh(d*x + c)) + (b^2*d*e - b^2*c*f)*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*e - b^2*c*f) 
*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*f*x + b^2*c*f)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*c 
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b^2*d*f*x 
 + b^2*c*f)*log(-(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) - ((a^2 + b^2)*d*f*x + (a^2 
+ b^2)*d*e)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (a^2*d*e + I*a*b*d*e 
- a^2*c*f - I*a*b*c*f)*log(cosh(d*x + c) + sinh(d*x + c) + I) + (a^2*d*e - 
 I*a*b*d*e - a^2*c*f + I*a*b*c*f)*log(cosh(d*x + c) + sinh(d*x + c) - I) - 
 ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*log(cosh(d*x + c) + sinh(d*x + c) - 1 
) + (a^2*d*f*x - I*a*b*d*f*x + a^2*c*f - I*a*b*c*f)*log(I*cosh(d*x + c) + 
I*sinh(d*x + c) + 1) + (a^2*d*f*x + I*a*b*d*f*x + a^2*c*f + I*a*b*c*f)*log 
(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^2 + b^2)*d*f*x + (a^2 + ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-e*(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) 
) + 4*f*integrate(2*x/((b*(e^(d*x + c) - e^(-d*x - c)) + 2*a)*(e^(d*x + c) 
 + e^(-d*x - c))*(e^(d*x + c) - e^(-d*x - c))), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((f*x+e)*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 + 8*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*f + 8*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*f - log(e**(2*c + 2*d*x) + 1)*a** 
2*e + log(e**(c + d*x) - 1)*a**2*e + log(e**(c + d*x) - 1)*b**2*e + log(e* 
*(c + d*x) + 1)*a**2*e + log(e**(c + d*x) + 1)*b**2*e - log(e**(2*c + 2*d* 
x)*b + 2*e**(c + d*x)*a - b)*b**2*e)/(a*d*(a**2 + b**2))