3.4.14 \(\int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [314]

3.4.14.1 Optimal result

Integrand size = 28, antiderivative size = 928 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {b^3 f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]

output

-2*I*a*b^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2+a*(f*x+e)^2* 
arctan(exp(d*x+c))/(a^2+b^2)/d-I*a*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2 
+b^2)/d^2-2*I*a*b^2*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)^2/d^3+1/2*b^3*f^ 
2*polylog(3,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^3+1/2*b*(f*x+e)^2*sech(d*x+c)^2 
/(a^2+b^2)/d-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b 
^2)^2/d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2) 
^2/d^3-a*f^2*arctan(sinh(d*x+c))/(a^2+b^2)/d^3-b^3*f*(f*x+e)*polylog(2,-ex 
p(2*d*x+2*c))/(a^2+b^2)^2/d^2+I*a*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d 
^3+a*f*(f*x+e)*sech(d*x+c)/(a^2+b^2)/d^2-b*f*(f*x+e)*tanh(d*x+c)/(a^2+b^2) 
/d^2-b^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d+b*f^2*ln(cosh(d*x+c) 
)/(a^2+b^2)/d^3+b^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+b^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*I*a*b^2*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)^2/d^3+I*a*f*(f*x+e)*pol 
ylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+2*I*a*b^2*f*(f*x+e)*polylog(2,I*exp(d*x 
+c))/(a^2+b^2)^2/d^2+2*a*b^2*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)^2/d+1/ 
2*a*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d+2*b^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*b^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*f^2*polylog(3, 
I*exp(d*x+c))/(a^2+b^2)/d^3
 

3.4.14.2 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. \(2(928)=1856\).

Time = 11.82 (sec) , antiderivative size = 3368, normalized size of antiderivative = 3.63 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

input

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

output

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

3.4.14.3 Rubi [A] (verified)

Time = 3.26 (sec) , antiderivative size = 764, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6107, 6107, 6095, 2620, 3011, 2720, 7143, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

(b^2*((b^2*(-1/3*(e + f*x)^3/(b*f) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/ 
(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + 
Sqrt[a^2 + b^2])])/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x)) 
/(a - Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a 
^2 + b^2]))])/d^2))/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x) 
)/(a + Sqrt[a^2 + b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[ 
a^2 + b^2]))])/d^2))/(b*d)))/(a^2 + b^2) + ((b*(e + f*x)^3)/(3*f) + (2*a*( 
e + f*x)^2*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)^2*Log[1 + E^(2*(c + d*x)) 
])/d - ((2*I)*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + ((2*I)*a*f 
*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*(e + f*x)*PolyLog[2, -E^( 
2*(c + d*x))])/d^2 + ((2*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/d^3 - ((2* 
I)*a*f^2*PolyLog[3, I*E^(c + d*x)])/d^3 + (b*f^2*PolyLog[3, -E^(2*(c + d*x 
))])/(2*d^3))/(a^2 + b^2)))/(a^2 + b^2) + ((a*(e + f*x)^2*ArcTan[E^(c + d* 
x)])/d - (a*f^2*ArcTan[Sinh[c + d*x]])/d^3 + (b*f^2*Log[Cosh[c + d*x]])/d^ 
3 - (I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*a*f*(e + f*x)* 
PolyLog[2, I*E^(c + d*x)])/d^2 + (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/d^ 
3 - (I*a*f^2*PolyLog[3, I*E^(c + d*x)])/d^3 + (a*f*(e + f*x)*Sech[c + d*x] 
)/d^2 + (b*(e + f*x)^2*Sech[c + d*x]^2)/(2*d) - (b*f*(e + f*x)*Tanh[c + d* 
x])/d^2 + (a*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*d))/(a^2 + b^2)
 

3.4.14.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2620

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

rule 2720

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

rule 3011

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

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6107

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2)   Int[(e + 
f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 
+ b^2)   Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 
]
 

rule 7143

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

rule 7293

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

3.4.14.4 Maple [F]

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

input

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

output

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

3.4.14.5 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. 10642 vs. \(2 (855) = 1710\).

Time = 0.44 (sec) , antiderivative size = 10642, normalized size of antiderivative = 11.47 \[ \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.4.14.6 Sympy [F]

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

input

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

output

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

3.4.14.7 Maxima [F]

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

input

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

output

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

3.4.14.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.4.14.9 Mupad [F(-1)]

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

input

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

output

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