Integrand size = 28, antiderivative size = 1479 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]
-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^2-1/2*(f*x+e)^2*sech(d*x+c)*tan h(d*x+c)/b/d-I*f^2*polylog(3,I*exp(d*x+c))/b/d^3+I*f*(f*x+e)*polylog(2,I*e xp(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-I*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b/d^2-2*a^4*(f*x+e)^2*arctan( 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-2*a^3*f*(f*x+e)*polylog( 2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-2*a^3*f*(f*x+e)*polyl og(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-I*a^2*f^2*polylog( 3,I*exp(d*x+c))/b^3/d^3+I*a^4*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^3/(a^2+ b^2)/d^2+2*I*a^4*f^2*polylog(3,I*exp(d*x+c))/b/(a^2+b^2)^2/d^3+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^2*f*(f*x+e)*polylog( 2,I*exp(d*x+c))/b^3/d^2+I*a^4*f^2*polylog(3,I*exp(d*x+c))/b^3/(a^2+b^2)/d^ 3-a^4*f*(f*x+e)*sech(d*x+c)/b^3/(a^2+b^2)/d^2+(f*x+e)^2*arctan(exp(d*x+c)) /b/d+f^2*arctan(sinh(d*x+c))/b/d^3-a^2*f^2*arctan(sinh(d*x+c))/b^3/d^3+a^4 *f^2*arctan(sinh(d*x+c))/b^3/(a^2+b^2)/d^3+a^3*f*(f*x+e)*polylog(2,-exp(2* d*x+2*c))/(a^2+b^2)^2/d^2+I*a^2*f^2*polylog(3,-I*exp(d*x+c))/b^3/d^3+a^2*f *(f*x+e)*sech(d*x+c)/b^3/d^2-a*f*(f*x+e)*tanh(d*x+c)/b^2/d^2-a^3*f^2*ln(co sh(d*x+c))/b^2/(a^2+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*f^2*ln(cosh(d*x+c))/b^2/d^3-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3368\) vs. \(2(1479)=2958\).
Time = 11.93 (sec) , antiderivative size = 3368, normalized size of antiderivative = 2.28 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]
(-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 +...
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}\) |
3.5.16.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]]]
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]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]*Tanh[(a_.) + (b_.)* (x_)]^(p_), x_Symbol] :> Int[(c + d*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2 ), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; FreeQ[ {a, b, c, d, m}, x] && IGtQ[p/2, 0]
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]
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 ]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(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*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*(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] && IGtQ[p, 0]
\[\int \frac {\left (f x +e \right )^{2} \tanh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]
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 = 0.46 (sec) , antiderivative size = 10574, normalized size of antiderivative = 7.15 \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
\[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \tanh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
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 + ...
Timed out. \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x)^2 \tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]