\(\int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [417]

Optimal result

Integrand size = 26, antiderivative size = 894 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

1/2*a^3*f*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2+a^3*(f*x+e)*ln(1+exp( 
2*d*x+2*c))/(a^2+b^2)^2/d+a^2*(f*x+e)*arctan(exp(d*x+c))/b^3/d-a^3*f*polyl 
og(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a^3*f*polylog(2,-b 
*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a^3*(f*x+e)*ln(1+b*exp(d* 
x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-( 
a^2+b^2)^(1/2)))/(a^2+b^2)^2/d-1/2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b/d-1/2 
*I*f*polylog(2,-I*exp(d*x+c))/b/d^2-a^4*(f*x+e)*arctan(exp(d*x+c))/b^3/(a^ 
2+b^2)/d-2*a^4*(f*x+e)*arctan(exp(d*x+c))/b/(a^2+b^2)^2/d+1/2*a^3*f*tanh(d 
*x+c)/b^2/(a^2+b^2)/d^2+1/2*a^2*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b^3/d-1/2* 
a^4*f*sech(d*x+c)/b^3/(a^2+b^2)/d^2-1/2*a^3*(f*x+e)*sech(d*x+c)^2/b^2/(a^2 
+b^2)/d-1/2*I*a^2*f*polylog(2,-I*exp(d*x+c))/b^3/d^2+1/2*I*f*polylog(2,I*e 
xp(d*x+c))/b/d^2+(f*x+e)*arctan(exp(d*x+c))/b/d-1/2*f*sech(d*x+c)/b/d^2-1/ 
2*a*f*tanh(d*x+c)/b^2/d^2+1/2*a^2*f*sech(d*x+c)/b^3/d^2+1/2*a*(f*x+e)*sech 
(d*x+c)^2/b^2/d+I*a^4*f*polylog(2,-I*exp(d*x+c))/b/(a^2+b^2)^2/d^2+1/2*I*a 
^4*f*polylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2+1/2*I*a^2*f*polylog(2,I*ex 
p(d*x+c))/b^3/d^2-1/2*a^4*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/b^3/(a^2+b^2)/d- 
I*a^4*f*polylog(2,I*exp(d*x+c))/b/(a^2+b^2)^2/d^2-1/2*I*a^4*f*polylog(2,I* 
exp(d*x+c))/b^3/(a^2+b^2)/d^2
 

Mathematica [A] (warning: unable to verify)

Time = 8.95 (sec) , antiderivative size = 834, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {-2 a^3 d e (c+d x)+2 a^3 c f (c+d x)-a^3 f (c+d x)^2+6 a^2 b d e \arctan \left (e^{c+d x}\right )+2 b^3 d e \arctan \left (e^{c+d x}\right )-6 a^2 b c f \arctan \left (e^{c+d x}\right )-2 b^3 c f \arctan \left (e^{c+d x}\right )+3 i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )+i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-3 i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (3 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (-b f-a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (a d e-a c f+a f (c+d x)-b d e \sinh (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \] Input:

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

Output:

-1/2*(a^3*(-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]))]))/((a^2 + b^2)^2*d^2) + (-2*a^3*d*e*(c + d*x) + 2*a^3*c*f*(c + 
 d*x) - a^3*f*(c + d*x)^2 + 6*a^2*b*d*e*ArcTan[E^(c + d*x)] + 2*b^3*d*e*Ar 
cTan[E^(c + d*x)] - 6*a^2*b*c*f*ArcTan[E^(c + d*x)] - 2*b^3*c*f*ArcTan[E^( 
c + d*x)] + (3*I)*a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + I*b^3*f*(c + 
d*x)*Log[1 - I*E^(c + d*x)] - (3*I)*a^2*b*f*(c + d*x)*Log[1 + I*E^(c + d*x 
)] - I*b^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^3*d*e*Log[1 + E^(2*(c 
+ d*x))] - 2*a^3*c*f*Log[1 + E^(2*(c + d*x))] + 2*a^3*f*(c + d*x)*Log[1 + 
E^(2*(c + d*x))] - I*b*(3*a^2 + b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b* 
(3*a^2 + b^2)*f*PolyLog[2, I*E^(c + d*x)] + a^3*f*PolyLog[2, -E^(2*(c + d* 
x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c + d*x]*(-(b*f) - a*f*Sinh[c + d*x])) 
/(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(a*d*e - a*c*f + a*f*(c + d*x) - b 
*d*e*Sinh[c + d*x] + b*c*f*Sinh[c + d*x] - b*f*(c + d*x)*Sinh[c + d*x])...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 6101

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

\(\Big \downarrow \) 5978

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6117

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4254

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6117

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

Input:

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

Output:

$Aborted
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (820 ) = 1640\).

Time = 2.47 (sec) , antiderivative size = 2284, normalized size of antiderivative = 2.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(2284\)

Input:

int((f*x+e)*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

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

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. 4729 vs. \(2 (793) = 1586\).

Time = 0.22 (sec) , antiderivative size = 4729, normalized size of antiderivative = 5.29 \[ \int \frac {(e+f x) \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral((e + f*x)*tanh(c + d*x)**3/(a + b*sinh(c + d*x)), x)
 

Maxima [F]

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

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

Output:

-(a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + 
b^4)*d) - a^3*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (3*a 
^2*b + b^3)*arctan(e^(-d*x - c))/((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))*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) 
) - integrate(-2*(a^4*x*e^(d*x + c) - a^3*b*x)/(a^4*b + 2*a^2*b^3 + b^5 - 
(a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 
 2*a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) - integrate(-(2*a^3*x - (3*a^2*b* 
e^c + b^3*e^c)*x*e^(d*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))
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(8*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**6*b*f + 24*e**(4*c + 4*d*x)*atan 
(e**(c + d*x))*a**4*b**3*f + 18*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b 
**5*d*e + 24*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b**5*f + 6*e**(4*c + 
 4*d*x)*atan(e**(c + d*x))*b**7*d*e + 8*e**(4*c + 4*d*x)*atan(e**(c + d*x) 
)*b**7*f + 16*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**6*b*f + 48*e**(2*c + 
2*d*x)*atan(e**(c + d*x))*a**4*b**3*f + 36*e**(2*c + 2*d*x)*atan(e**(c + d 
*x))*a**2*b**5*d*e + 48*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**5*f + 
12*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**7*d*e + 16*e**(2*c + 2*d*x)*atan 
(e**(c + d*x))*b**7*f + 8*atan(e**(c + d*x))*a**6*b*f + 24*atan(e**(c + d* 
x))*a**4*b**3*f + 18*atan(e**(c + d*x))*a**2*b**5*d*e + 24*atan(e**(c + d* 
x))*a**2*b**5*f + 6*atan(e**(c + d*x))*b**7*d*e + 8*atan(e**(c + d*x))*b** 
7*f + 192*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**( 
7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5*d*x)*a + 6*e**(3*c + 
 3*d*x)*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**8*d**2*f + 
400*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 
7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5*d*x)*a + 6*e**(3*c + 3*d*x 
)*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**6*b**2*d**2*f + 2 
40*e**(7*c + 4*d*x)*int((e**(3*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 7 
*d*x)*a + 2*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5*d*x)*a + 6*e**(3*c + 3*d*x) 
*a - 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**4*d**2*f +...