\(\int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [416]

Optimal result

Integrand size = 28, antiderivative size = 1479 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

I*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b/d^2+I*f^2*polylog(3,-I*exp(d*x+c))/b 
/d^3-f*(f*x+e)*sech(d*x+c)/b/d^2+f^2*arctan(sinh(d*x+c))/b/d^3+a*f^2*ln(co 
sh(d*x+c))/b^2/d^3+a^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d-a^2*f^ 
2*arctan(sinh(d*x+c))/b^3/d^3+a^3*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/(a^ 
2+b^2)^2/d^2+a^4*f^2*arctan(sinh(d*x+c))/b^3/(a^2+b^2)/d^3-a^3*f^2*ln(cosh 
(d*x+c))/b^2/(a^2+b^2)/d^3-a*f*(f*x+e)*tanh(d*x+c)/b^2/d^2+a^2*f*(f*x+e)*s 
ech(d*x+c)/b^3/d^2+I*a^2*f^2*polylog(3,-I*exp(d*x+c))/b^3/d^3-1/2*a^4*(f*x 
+e)^2*sech(d*x+c)*tanh(d*x+c)/b^3/(a^2+b^2)/d-2*I*a^4*f^2*polylog(3,-I*exp 
(d*x+c))/b/(a^2+b^2)^2/d^3-I*a^4*f^2*polylog(3,-I*exp(d*x+c))/b^3/(a^2+b^2 
)/d^3-I*a^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^3/d^2+a^2*(f*x+e)^2*arcta 
n(exp(d*x+c))/b^3/d-2*I*a^4*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b/(a^2+b^2)^ 
2/d^2-I*a^4*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-a^3*(f*x+e 
)^2*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)^2*ln( 
1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+2*a^3*f^2*polylog(3,-b*e 
xp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^3+2*a^3*f^2*polylog(3,-b*exp( 
d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^3+(f*x+e)^2*arctan(exp(d*x+c))/b 
/d-1/2*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/b/d-I*f^2*polylog(3,I*exp(d*x+c)) 
/b/d^3-a^4*(f*x+e)^2*arctan(exp(d*x+c))/b^3/(a^2+b^2)/d-2*a^4*(f*x+e)^2*ar 
ctan(exp(d*x+c))/b/(a^2+b^2)^2/d-1/2*a^3*(f*x+e)^2*sech(d*x+c)^2/b^2/(a^2+ 
b^2)/d+1/2*a^2*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/b^3/d-I*a^2*f^2*polylo...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3368\) vs. \(2(1479)=2958\).

Time = 11.47 (sec) , antiderivative size = 3368, normalized size of antiderivative = 2.28 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

Integrate[((e + f*x)^2*Tanh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
 

Output:

(-12*a^3*d^3*e^2*E^(2*c)*x - 12*a^3*d*E^(2*c)*f^2*x - 12*a*b^2*d*E^(2*c)*f 
^2*x - 12*a^3*d^3*e*E^(2*c)*f*x^2 - 4*a^3*d^3*E^(2*c)*f^2*x^3 + 18*a^2*b*d 
^2*e^2*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*ArcTan[E^(c + d*x)] + 18*a^2*b* 
d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + 
d*x)] + 12*a^2*b*f^2*ArcTan[E^(c + d*x)] + 12*b^3*f^2*ArcTan[E^(c + d*x)] 
+ 12*a^2*b*E^(2*c)*f^2*ArcTan[E^(c + d*x)] + 12*b^3*E^(2*c)*f^2*ArcTan[E^( 
c + d*x)] + (18*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b^3*d^2* 
e*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^ 
(c + d*x)] + (6*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (9*I)*a^ 
2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*d^2*f^2*x^2*Log[1 - I*E 
^(c + d*x)] + (9*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3* 
I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (18*I)*a^2*b*d^2*e*f*x 
*Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (18 
*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*E^(2* 
c)*f*x*Log[1 + I*E^(c + d*x)] - (9*I)*a^2*b*d^2*f^2*x^2*Log[1 + I*E^(c + d 
*x)] - (3*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I)*a^2*b*d^2*E^(2 
*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (3*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + 
 I*E^(c + d*x)] + 6*a^3*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*e^2*E 
^(2*c)*Log[1 + E^(2*(c + d*x))] + 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] + 6*a 
*b^2*f^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*E^(2*c)*f^2*Log[1 + E^(2*(c +...
 

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.

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

\(\Big \downarrow \) 6101

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

\(\Big \downarrow \) 5978

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4674

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4257

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6117

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

\(\Big \downarrow \) 5974

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 6117

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4674

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4257

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6107

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

Input:

Int[((e + f*x)^2*Tanh[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \tanh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]

Input:

int((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
 

Output:

int((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
 

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. 10574 vs. \(2 (1354) = 2708\).

Time = 0.30 (sec) , antiderivative size = 10574, normalized size of antiderivative = 7.15 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)**2*tanh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
 

Output:

Integral((e + f*x)**2*tanh(c + d*x)**3/(a + b*sinh(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \tanh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:

int((tanh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)
 

Output:

int((tanh(c + d*x)^3*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)^2*tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
 

Output:

(48*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**6*b*d*e*f + 16*e**(4*c + 4*d*x) 
*atan(e**(c + d*x))*a**6*b*f**2 + 144*e**(4*c + 4*d*x)*atan(e**(c + d*x))* 
a**4*b**3*d*e*f + 120*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**4*b**3*f**2 + 
 54*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b**5*d**2*e**2 + 144*e**(4*c 
+ 4*d*x)*atan(e**(c + d*x))*a**2*b**5*d*e*f + 192*e**(4*c + 4*d*x)*atan(e* 
*(c + d*x))*a**2*b**5*f**2 + 18*e**(4*c + 4*d*x)*atan(e**(c + d*x))*b**7*d 
**2*e**2 + 48*e**(4*c + 4*d*x)*atan(e**(c + d*x))*b**7*d*e*f + 88*e**(4*c 
+ 4*d*x)*atan(e**(c + d*x))*b**7*f**2 + 96*e**(2*c + 2*d*x)*atan(e**(c + d 
*x))*a**6*b*d*e*f + 32*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**6*b*f**2 + 2 
88*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4*b**3*d*e*f + 240*e**(2*c + 2*d 
*x)*atan(e**(c + d*x))*a**4*b**3*f**2 + 108*e**(2*c + 2*d*x)*atan(e**(c + 
d*x))*a**2*b**5*d**2*e**2 + 288*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b 
**5*d*e*f + 384*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b**5*f**2 + 36*e* 
*(2*c + 2*d*x)*atan(e**(c + d*x))*b**7*d**2*e**2 + 96*e**(2*c + 2*d*x)*ata 
n(e**(c + d*x))*b**7*d*e*f + 176*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**7* 
f**2 + 48*atan(e**(c + d*x))*a**6*b*d*e*f + 16*atan(e**(c + d*x))*a**6*b*f 
**2 + 144*atan(e**(c + d*x))*a**4*b**3*d*e*f + 120*atan(e**(c + d*x))*a**4 
*b**3*f**2 + 54*atan(e**(c + d*x))*a**2*b**5*d**2*e**2 + 144*atan(e**(c + 
d*x))*a**2*b**5*d*e*f + 192*atan(e**(c + d*x))*a**2*b**5*f**2 + 18*atan(e* 
*(c + d*x))*b**7*d**2*e**2 + 48*atan(e**(c + d*x))*b**7*d*e*f + 88*atan...