\(\int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [491]

Optimal result

Integrand size = 34, antiderivative size = 1795 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}+\frac {2 b (e+f x)^3 \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {6 b f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {b^4 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {3 i b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {3 i b^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 b^4 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 b^4 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {6 b f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a^2 d^3}-\frac {6 i b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a^2 d^3}+\frac {6 i b^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^4 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 b^4 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 b^4 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {6 i b f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a^2 d^4}+\frac {6 i b^3 f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {6 i b f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{a^2 d^4}-\frac {6 i b^3 f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 b^4 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 b^4 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 a^3 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 a d^4}-\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 a^3 d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 a d^4}+\frac {3 b^2 f^3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 a^3 d^4} \]

[Out]

2*b*(f*x+e)^3*arctan(exp(d*x+c))/a^2/d-2*b^2*(f*x+e)^3*arctanh(exp(2*d*x+2*c))/a^3/d-3/4*b^2*f^3*polylog(4,-ex
p(2*d*x+2*c))/a^3/d^4+6*I*b*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/a^2/d^3+6*I*b^3*f^3*polylog(4,-I*exp(d*x+c))/
a^2/(a^2+b^2)/d^4+3*I*b*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a^2/d^2+3*I*b^3*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c
))/a^2/(a^2+b^2)/d^2-3/2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a/d^2+3/2*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))
/a/d^3-3/2*f*(f*x+e)^2*coth(d*x+c)/a/d^2+3*f^2*(f*x+e)*ln(1-exp(2*d*x+2*c))/a/d^3-6*b*f^3*polylog(3,-exp(d*x+c
))/a^2/d^4+6*b*f^3*polylog(3,exp(d*x+c))/a^2/d^4+3/4*b^2*f^3*polylog(4,exp(2*d*x+2*c))/a^3/d^4-3/2*f*(f*x+e)^2
/a/d^2-6*I*b^3*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^3-3*I*b^3*f*(f*x+e)^2*polylog(2,I*exp(d*x+
c))/a^2/(a^2+b^2)/d^2+3/2*f^3*polylog(2,exp(2*d*x+2*c))/a/d^4-1/2*(f*x+e)^3*coth(d*x+c)^2/a/d+1/2*(f*x+e)^3/a/
d+2*(f*x+e)^3*arctanh(exp(2*d*x+2*c))/a/d+3/4*f^3*polylog(4,-exp(2*d*x+2*c))/a/d^4+6*b*f*(f*x+e)^2*arctanh(exp
(d*x+c))/a^2/d^2+6*b*f^2*(f*x+e)*polylog(2,-exp(d*x+c))/a^2/d^3-6*b*f^2*(f*x+e)*polylog(2,exp(d*x+c))/a^2/d^3+
3/2*b^2*f*(f*x+e)^2*polylog(2,exp(2*d*x+2*c))/a^3/d^2-3/2*b^2*f^2*(f*x+e)*polylog(3,exp(2*d*x+2*c))/a^3/d^3+3/
2*b^4*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^2-3/2*b^4*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/
a^3/(a^2+b^2)/d^3-3*b^4*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2-3*b^4*f*(f*
x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2+6*b^4*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c
)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^3+6*b^4*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a
^2+b^2)/d^3-3*I*b*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/a^2/d^2-6*I*b*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/a^2/d
^3-6*I*b^3*f^3*polylog(4,I*exp(d*x+c))/a^2/(a^2+b^2)/d^4-2*b^3*(f*x+e)^3*arctan(exp(d*x+c))/a^2/(a^2+b^2)/d-3/
2*b^2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+3/2*b^2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/a^3/d^3+3/
4*b^4*f^3*polylog(4,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^4-6*b^4*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/
a^3/(a^2+b^2)/d^4-6*b^4*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^4-6*I*b*f^3*polylog(4
,-I*exp(d*x+c))/a^2/d^4+b^4*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b^2)/d-b^4*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-
(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-3/4*f^3
*polylog(4,exp(2*d*x+2*c))/a/d^4+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/a/d^2-3/2*f^2*(f*x+e)*polylog(3,-e
xp(2*d*x+2*c))/a/d^3+b*(f*x+e)^3*csch(d*x+c)/a^2/d+6*I*b^3*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/a^2/(a^2+b^2)/d
^3+6*I*b*f^3*polylog(4,I*exp(d*x+c))/a^2/d^4

Rubi [A] (verified)

Time = 2.27 (sec) , antiderivative size = 1795, normalized size of antiderivative = 1.00, number of steps used = 87, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {5708, 2700, 14, 5570, 6873, 12, 6874, 3801, 3797, 2221, 2317, 2438, 32, 2631, 4267, 2611, 6744, 2320, 6724, 2701, 327, 213, 5313, 4265, 5569, 5692, 5680, 3799} \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right ) b^4}{4 a^3 \left (a^2+b^2\right ) d^4}-\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^4}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) b^2}{2 a^3 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right ) b^2}{4 a^3 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right ) b^2}{4 a^3 d^4}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right ) b}{a^2 d}+\frac {6 f (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {(e+f x)^3 \text {csch}(c+d x) b}{a^2 d}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) b}{a^2 d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b}{a^2 d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b}{a^2 d^2}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) b}{a^2 d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) b}{a^2 d^4}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) b}{a^2 d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) b}{a^2 d^3}+\frac {6 f^3 \operatorname {PolyLog}\left (3,e^{c+d x}\right ) b}{a^2 d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right ) b}{a^2 d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right ) b}{a^2 d^4}+\frac {(e+f x)^3}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (4,-e^{2 c+2 d x}\right )}{4 a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (4,e^{2 c+2 d x}\right )}{4 a d^4} \]

[In]

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

[Out]

(-3*f*(e + f*x)^2)/(2*a*d^2) + (e + f*x)^3/(2*a*d) + (2*b*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e
+ f*x)^3*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (2*(
e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) - (3*f*(e +
f*x)^2*Coth[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d
) - (b^4*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)^3*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])
/(a*d^3) + (b^4*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (6*b*f^2*(e + f*x)*PolyLog[2, -E^(
c + d*x)])/(a^2*d^3) - ((3*I)*b*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^2) + ((3*I)*b^3*f*(e + f*x)
^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a^
2*d^2) - ((3*I)*b^3*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (6*b*f^2*(e + f*x)*PolyLo
g[2, E^(c + d*x)])/(a^2*d^3) - (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3
*(a^2 + b^2)*d^2) - (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2
)*d^2) + (3*b^4*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^2) + (3*f^3*PolyLog[2, E^(2*(
c + d*x))])/(2*a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*PolyLo
g[2, -E^(2*c + 2*d*x)])/(2*a^3*d^2) - (3*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) + (3*b^2*f*(e +
f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^3*d^2) - (6*b*f^3*PolyLog[3, -E^(c + d*x)])/(a^2*d^4) + ((6*I)*b*f^2*
(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - ((6*I)*b^3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a^
2*(a^2 + b^2)*d^3) - ((6*I)*b*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + ((6*I)*b^3*f^2*(e + f*x)*Po
lyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*b^4*f^2*(e +
f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) + (6*b^4*f^2*(e + f*x)*PolyL
og[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) - (3*b^4*f^2*(e + f*x)*PolyLog[3, -E^(2
*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) + (3*b^2*f^2*
(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3)
- (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^3*d^3) - ((6*I)*b*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(
a^2*d^4) + ((6*I)*b^3*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) + ((6*I)*b*f^3*PolyLog[4, I*E^(c
+ d*x)])/(a^2*d^4) - ((6*I)*b^3*f^3*PolyLog[4, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) - (6*b^4*f^3*PolyLog[4,
-((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^4) - (6*b^4*f^3*PolyLog[4, -((b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^4) + (3*b^4*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a^3*(a^2 + b^2)*d^4)
+ (3*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a*d^4) - (3*b^2*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a^3*d^4) - (3*
f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a*d^4) + (3*b^2*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a^3*d^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

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] + Dist[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 2631

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1)*(Log[u]/(b*(m + 1))), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[(a + b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 4265

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] + (-Dist[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] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5313

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan[
u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 + u^2)), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(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 5692

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

Rule 5708

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x)^3 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(3 f) \int (e+f x)^2 \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a} \\ & = \frac {b (e+f x)^3 \arctan (\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x)^3 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(3 f) \int \frac {(e+f x)^2 \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right )}{2 d} \, dx}{a}+\frac {(3 b f) \int (e+f x)^2 \left (-\frac {\arctan (\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2} \\ & = \frac {b (e+f x)^3 \arctan (\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 b f) \int \frac {(e+f x)^2 (-\arctan (\sinh (c+d x))-\text {csch}(c+d x))}{d} \, dx}{a^2}-\frac {(3 f) \int (e+f x)^2 \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right ) \, dx}{2 a d} \\ & = \frac {b^4 (e+f x)^4}{4 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^3 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {b^3 \int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {(3 f) \int \left (-(e+f x)^2 \coth ^2(c+d x)-2 (e+f x)^2 \log (\tanh (c+d x))\right ) \, dx}{2 a d}+\frac {(3 b f) \int (e+f x)^2 (-\arctan (\sinh (c+d x))-\text {csch}(c+d x)) \, dx}{a^2 d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d} \\ & = \frac {b^4 (e+f x)^4}{4 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^3 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log (\tanh (c+d x))}{a d}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {b^3 \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x)^3 \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 f) \int (e+f x)^2 \coth ^2(c+d x) \, dx}{2 a d}+\frac {(3 f) \int (e+f x)^2 \log (\tanh (c+d x)) \, dx}{a d}+\frac {(3 b f) \int \left (-(e+f x)^2 \arctan (\sinh (c+d x))-(e+f x)^2 \text {csch}(c+d x)\right ) \, dx}{a^2 d}+\frac {\left (3 b^4 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (3 b^4 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (3 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}-\frac {\left (3 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) \, dx}{a^3 d^2} \\ & = -\frac {2 b^3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^3 \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {3 b^4 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^4 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {3 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\int 2 d (e+f x)^3 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \, dx}{2 a d}-\frac {(3 b f) \int (e+f x)^2 \arctan (\sinh (c+d x)) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^2 d}+\frac {\left (3 i b^3 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (3 i b^3 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (3 f^2\right ) \int (e+f x) \coth (c+d x) \, dx}{a d^2}+\frac {\left (6 b^4 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (6 b^4 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}-\frac {\left (3 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right ) \, dx}{2 a^3 d^3}+\frac {\left (3 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right ) \, dx}{2 a^3 d^3} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5813\) vs. \(2(1795)=3590\).

Time = 12.84 (sec) , antiderivative size = 5813, normalized size of antiderivative = 3.24 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^3*csch(d*x+c)^3*sech(d*x+c)/(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. 23903 vs. \(2 (1636) = 3272\).

Time = 0.77 (sec) , antiderivative size = 23903, normalized size of antiderivative = 13.32 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b
^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x -
3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) +
(a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^3 + (3*a*f^3*x^2 + 6*a*e*f^2*x + 3*a*e^2*f + 2*(b*d*f^3*x^3*e^(3*
c) + 3*b*d*e*f^2*x^2*e^(3*c) + 3*b*d*e^2*f*x*e^(3*c))*e^(3*d*x) - (2*a*d*f^3*x^3*e^(2*c) + 3*a*e^2*f*e^(2*c) +
3*(2*d*e*f^2 + f^3)*a*x^2*e^(2*c) + 6*(d*e^2*f + e*f^2)*a*x*e^(2*c))*e^(2*d*x) - 2*(b*d*f^3*x^3*e^c + 3*b*d*e
*f^2*x^2*e^c + 3*b*d*e^2*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) - 3
*(b*d*e^2*f + a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f - a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f + a*e*f^2)*log(e^(d*x
+ c) + 1)/(a^2*d^3) - 3*(b*d*e^2*f - a*e*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) - (d^3*x^3*log(e^(d*x + c) + 1) +
3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*(a^2*f^3 - b^2*f
^3)/(a^3*d^4) - (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c))
+ 6*polylog(4, e^(d*x + c)))*(a^2*f^3 - b^2*f^3)/(a^3*d^4) - 3*(a^2*d*e*f^2 - b^2*d*e*f^2 - a*b*f^3)*(d^2*x^2*
log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^3*d^4) - 3*(a^2*d*e*f^2 - b^
2*d*e*f^2 + a*b*f^3)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))/(a
^3*d^4) + 3*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 - (d^2*e^2*f - f^3)*a^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x
+ c)))/(a^3*d^4) + 3*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 - (d^2*e^2*f - f^3)*a^2)*(d*x*log(-e^(d*x + c) + 1) + dilo
g(e^(d*x + c)))/(a^3*d^4) + 1/4*((a^2*f^3 - b^2*f^3)*d^4*x^4 + 4*(a^2*d*e*f^2 - b^2*d*e*f^2 + a*b*f^3)*d^3*x^3
- 6*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 - (d^2*e^2*f - f^3)*a^2)*d^2*x^2)/(a^3*d^4) + 1/4*((a^2*f^3 - b^2*f^3)*d^4
*x^4 + 4*(a^2*d*e*f^2 - b^2*d*e*f^2 - a*b*f^3)*d^3*x^3 - 6*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 - (d^2*e^2*f - f^3)*
a^2)*d^2*x^2)/(a^3*d^4) + integrate(2*(b^5*f^3*x^3 + 3*b^5*e*f^2*x^2 + 3*b^5*e^2*f*x - (a*b^4*f^3*x^3*e^c + 3*
a*b^4*e*f^2*x^2*e^c + 3*a*b^4*e^2*f*x*e^c)*e^(d*x))/(a^5*b + a^3*b^3 - (a^5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*
d*x) - 2*(a^6*e^c + a^4*b^2*e^c)*e^(d*x)), x) + integrate(-2*(a*f^3*x^3 + 3*a*e*f^2*x^2 + 3*a*e^2*f*x - (b*f^3
*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)),
x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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