3.4.77 \(\int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [377]

3.4.77.1 Optimal result

Integrand size = 32, antiderivative size = 1218 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^4}{4 b f}-\frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \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)^3 \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)^3 \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {3 i a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {3 i a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 a^2 f (e+f x)^2 \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 {3 a^2 f (e+f x)^2 \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 {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {3 a^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^3}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^3}+\frac {3 a^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^3}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{b^2 d^4}-\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b d^4}-\frac {3 a^2 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b \left (a^2+b^2\right ) d^4} \]

output

a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d-6*I*a^3 
*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b^2/(a^2+b^2)/d^3-3*I*a^3*f*(f*x+e)^2 
*polylog(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+(f*x+e)^3*ln(1+exp(2*d*x+2*c)) 
/b/d-3/2*a^2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2+3/2*a^ 
2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^3+3*a^2*f*(f*x+e)^2 
*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2+3*a^2*f*(f*x 
+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2-6*a^2*f 
^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^3+6* 
I*a*f^3*polylog(4,-I*exp(d*x+c))/b^2/d^4-1/4*(f*x+e)^4/b/f-6*a^2*f^2*(f*x+ 
e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^3-3*I*a*f*(f 
*x+e)^2*polylog(2,I*exp(d*x+c))/b^2/d^2-6*I*a*f^2*(f*x+e)*polylog(3,-I*exp 
(d*x+c))/b^2/d^3-6*I*a^3*f^3*polylog(4,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+3/ 
4*f^3*polylog(4,-exp(2*d*x+2*c))/b/d^4-2*a*(f*x+e)^3*arctan(exp(d*x+c))/b^ 
2/d+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/d^2-3/2*f^2*(f*x+e)*polyl 
og(3,-exp(2*d*x+2*c))/b/d^3+3*I*a*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^2 
/d^2+6*I*a*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b^2/d^3+6*I*a^3*f^3*polylog 
(4,I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+3*I*a^3*f*(f*x+e)^2*polylog(2,I*exp(d*x 
+c))/b^2/(a^2+b^2)/d^2+6*I*a^3*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^2/(a 
^2+b^2)/d^3+2*a^3*(f*x+e)^3*arctan(exp(d*x+c))/b^2/(a^2+b^2)/d-3/4*a^2*f^3 
*polylog(4,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^4+6*a^2*f^3*polylog(4,-b*exp(...
 

3.4.77.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3251\) vs. \(2(1218)=2436\).

Time = 11.15 (sec) , antiderivative size = 3251, normalized size of antiderivative = 2.67 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

input

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

output

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

3.4.77.3 Rubi [A] (verified)

Time = 5.14 (sec) , antiderivative size = 1036, normalized size of antiderivative = 0.85, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {6115, 3042, 26, 4201, 2620, 3011, 6101, 3042, 4668, 3011, 6107, 6095, 2620, 3011, 7163, 2720, 7143, 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.

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

\(\Big \downarrow \) 6115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)}dx}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 6101

\(\displaystyle -\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 4668

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 6095

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}+\frac {\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \int \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )dx}{2 d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}-\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {a \left (-\frac {a \left (\frac {\int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right )dx}{a^2+b^2}+\frac {b^2 \left (-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^4}{4 b f}\right )}{a^2+b^2}\right )}{b}+\frac {\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (\frac {\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}}{b}-\frac {a \left (\frac {\left (-\frac {(e+f x)^4}{4 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^4}{4 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)^3}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{d}-\frac {3 i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) (e+f x)^2}{d^2}+\frac {3 i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) (e+f x)^2}{d^2}-\frac {3 b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) (e+f x)^2}{2 d^2}+\frac {6 i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) (e+f x)}{d^3}-\frac {6 i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) (e+f x)}{d^3}+\frac {3 b f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) (e+f x)}{2 d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^4}}{a^2+b^2}\right )}{b}\right )}{b}-\frac {i \left (2 i \left (\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{2 d}-\frac {3 f \left (\frac {f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}\right )}{2 d}\right )-\frac {i (e+f x)^4}{4 f}\right )}{b}\)

input

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

output

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

3.4.77.3.1 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.4.77.4 Maple [F]

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

input

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

output

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

3.4.77.5 Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1962, normalized size of antiderivative = 1.61 \[ \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

-1/4*((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*e*f^2*x^3 + 6*(a^2 + b^2 
)*d^4*e^2*f*x^2 + 4*(a^2 + b^2)*d^4*e^3*x - 24*a^2*f^3*polylog(4, (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) - 24*a^2*f^3*polylog(4, (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) - 12*(a^ 
2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^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) - 12*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*e*f^2*x + a^2*d^2*e^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) + 12*(I*a*b*d^2*f^3*x^2 - b^2*d^2* 
f^3*x^2 + 2*I*a*b*d^2*e*f^2*x - 2*b^2*d^2*e*f^2*x + I*a*b*d^2*e^2*f - b^2* 
d^2*e^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 12*(-I*a*b*d^2*f^3*x 
^2 - b^2*d^2*f^3*x^2 - 2*I*a*b*d^2*e*f^2*x - 2*b^2*d^2*e*f^2*x - I*a*b*d^2 
*e^2*f - b^2*d^2*e^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 4*(a^2 
*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*log(2*b*co 
sh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(a^ 
2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3)*log(2*b*c 
osh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(a 
^2*d^3*f^3*x^3 + 3*a^2*d^3*e*f^2*x^2 + 3*a^2*d^3*e^2*f*x + 3*a^2*c*d^2*e^2 
*f - 3*a^2*c^2*d*e*f^2 + a^2*c^3*f^3)*log(-(a*cosh(d*x + c) + a*sinh(d*...
 

3.4.77.6 Sympy [F]

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

input

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

output

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

3.4.77.7 Maxima [F]

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

input

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

output

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

3.4.77.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.4.77.9 Mupad [F(-1)]

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

input

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

output

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