3.4.79 \(\int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [379]

3.4.79.1 Optimal result

Integrand size = 30, antiderivative size = 516 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2}{2 b f}-\frac {2 a (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x) \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {i a^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {i a^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2} \]

output

-1/2*(f*x+e)^2/b/f-2*a*(f*x+e)*arctan(exp(d*x+c))/b^2/d+2*a^3*(f*x+e)*arct 
an(exp(d*x+c))/b^2/(a^2+b^2)/d+(f*x+e)*ln(1+exp(2*d*x+2*c))/b/d-a^2*(f*x+e 
)*ln(1+exp(2*d*x+2*c))/b/(a^2+b^2)/d+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2 
+b^2)^(1/2)))/b/(a^2+b^2)/d+a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/ 
2)))/b/(a^2+b^2)/d+I*a*f*polylog(2,-I*exp(d*x+c))/b^2/d^2-I*a^3*f*polylog( 
2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^2-I*a*f*polylog(2,I*exp(d*x+c))/b^2/d^2+I 
*a^3*f*polylog(2,I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+1/2*f*polylog(2,-exp(2*d* 
x+2*c))/b/d^2-1/2*a^2*f*polylog(2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2+a^2*f*p 
olylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2+a^2*f*polylog( 
2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2
 

3.4.79.2 Mathematica [A] (verified)

Time = 3.92 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.12 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 b d e (c+d x)+2 b c f (c+d x)-b f (c+d x)^2-4 a d e \arctan \left (e^{c+d x}\right )+4 a c f \arctan \left (e^{c+d x}\right )-2 i a f (c+d x) \log \left (1-i e^{c+d x}\right )+2 i a f (c+d x) \log \left (1+i e^{c+d x}\right )+2 b d e \log \left (1+e^{2 (c+d x)}\right )-2 b c f \log \left (1+e^{2 (c+d x)}\right )+2 b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+2 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-2 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {a^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b}+b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2} \]

input

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

output

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

3.4.79.3 Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.91, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.633, Rules used = {6115, 3042, 26, 4201, 2620, 2715, 2838, 6101, 3042, 4668, 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.

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

\(\Big \downarrow \) 6115

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int -i (e+f x) \tan (i c+i d x)dx}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {a \int \frac {(e+f x) \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \int (e+f x) \tan (i c+i d x)dx}{b}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6101

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.4.79.3.1 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 4201

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

rule 4668

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

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 6101

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b   Int[(e + f*x)^m*Sech[ 
c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b   Int[(e + f*x)^m*Sech[c 
+ d*x]*(Tanh[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]
 

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 6115

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + 
 (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/b   Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x] - S 
imp[a/b   Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(a + b*Sin 
h[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] 
]
 

3.4.79.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3881 vs. \(2 (484 ) = 968\).

Time = 2.07 (sec) , antiderivative size = 3882, normalized size of antiderivative = 7.52

method	result	size
risch	\(\text {Expression too large to display}\)	\(3882\)

input

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

output

2/d*e/(2*a^2+2*b^2)*b*ln(1+exp(2*d*x+2*c))-4/d*e/(2*a^2+2*b^2)*a*arctan(ex 
p(d*x+c))-1/d*e*b/(2*a^2+2*b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/b/ 
d^2*f/(a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)) 
)*a^2+1/b/d^2*f/(a^2+b^2)^(3/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(- 
a+(a^2+b^2)^(1/2)))*a^3-1/b/d^2*f/(a^2+b^2)^(3/2)*dilog((b*exp(d*x+c)+(a^2 
+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^3+1/b/d*e/(a^2+b^2)*ln(b*exp(2*d*x+2 
*c)+2*a*exp(d*x+c)-b)*a^2-2/b/d*e/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x 
+c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/d^2/(a^2+b^2)^(1/2)*a*b*c*f/(2*a^2+2*b^2)* 
arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/b/d^2*f/(a^2+b^2)*ln(( 
b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^2*c+2/b/d^2*a^3*f/( 
2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^ 
2+b^2)^(1/2)))-2/b/d^2*a^3*f/(2*a^2+2*b^2)/(a^2+b^2)^(1/2)*dilog((-b*exp(d 
*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/b/d*f/(a^2+b^2)*ln((-b*ex 
p(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2*x+1/b/d^2*f/(a^2+b^2 
)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a^2*c+1/b/d*f 
/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*a^2*x+ 
1/b/d*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2) 
^(1/2)))*a^3*x+1/b/d^2*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2) 
-a)/(-a+(a^2+b^2)^(1/2)))*a^3*c+b/d*f/(a^2+b^2)^(3/2)*ln((-b*exp(d*x+c)+(a 
^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*a*x+b/d^2*f/(a^2+b^2)^(3/2)*ln((...
 

3.4.79.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.32 \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (a^{2} + b^{2}\right )} d^{2} f x^{2} + 2 \, {\left (a^{2} + b^{2}\right )} d^{2} e x - 2 \, a^{2} f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 \, a^{2} f {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, {\left (i \, a b f - b^{2} f\right )} {\rm Li}_2\left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right )\right ) + 2 \, {\left (-i \, a b f - b^{2} f\right )} {\rm Li}_2\left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right )\right ) - 2 \, {\left (a^{2} d e - a^{2} c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (a^{2} d e - a^{2} c f\right )} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (i \, a b d e - b^{2} d e - i \, a b c f + b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + i\right ) + 2 \, {\left (-i \, a b d e - b^{2} d e + i \, a b c f + b^{2} c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - i\right ) + 2 \, {\left (-i \, a b d f x - b^{2} d f x - i \, a b c f - b^{2} c f\right )} \log \left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (i \, a b d f x - b^{2} d f x + i \, a b c f - b^{2} c f\right )} \log \left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d^{2}} \]

input

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

output

-1/2*((a^2 + b^2)*d^2*f*x^2 + 2*(a^2 + b^2)*d^2*e*x - 2*a^2*f*dilog((a*cos 
h(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) - 2*a^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 
) + 2*(I*a*b*f - b^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 2*(-I*a 
*b*f - b^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*(a^2*d*e - a^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) - 2*(a^2*d*e - a^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) - 2*(a^2*d*f*x + a^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) - 2*(a^2*d*f*x + a^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) + 2*(I*a*b*d*e - b^2*d*e - I*a*b*c*f + b^2*c*f)*log(cosh(d*x + c) 
+ sinh(d*x + c) + I) + 2*(-I*a*b*d*e - b^2*d*e + I*a*b*c*f + b^2*c*f)*log( 
cosh(d*x + c) + sinh(d*x + c) - I) + 2*(-I*a*b*d*f*x - b^2*d*f*x - I*a*b*c 
*f - b^2*c*f)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + 2*(I*a*b*d*f*x 
- b^2*d*f*x + I*a*b*c*f - b^2*c*f)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) 
+ 1))/((a^2*b + b^3)*d^2)
 

3.4.79.6 Sympy [F]

\[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \sinh {\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

input

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

output

Integral((e + f*x)*sinh(c + d*x)*tanh(c + d*x)/(a + b*sinh(c + d*x)), x)
 

3.4.79.7 Maxima [F]

\[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sinh \left (d x + c\right ) \tanh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]

input

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

output

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

3.4.79.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.4.79.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )\,\mathrm {tanh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

input

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

output

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