Integrand size = 18, antiderivative size = 326 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=-\frac {3}{2} x^2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} x^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x^3 \text {arctanh}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {3}{2} \cosh ^2(x) \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {3}{2} x^2 \cosh ^2(x) \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {3}{2} x^2 \cosh ^2(x) \operatorname {PolyLog}\left (2,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {3}{2} x \cosh ^2(x) \operatorname {PolyLog}\left (3,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {3}{2} x \cosh ^2(x) \operatorname {PolyLog}\left (3,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {3}{4} \cosh ^2(x) \operatorname {PolyLog}\left (4,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {3}{4} \cosh ^2(x) \operatorname {PolyLog}\left (4,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {3}{2} x^2 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \]
-3/2*x^2*cosh(x)^2*(a*sech(x)^4)^(1/2)+1/2*x^3*cosh(x)^2*(a*sech(x)^4)^(1/ 2)-2*x^3*arctanh(exp(2*x))*cosh(x)^2*(a*sech(x)^4)^(1/2)+3*x*cosh(x)^2*ln( 1+exp(2*x))*(a*sech(x)^4)^(1/2)+3/2*cosh(x)^2*polylog(2,-exp(2*x))*(a*sech (x)^4)^(1/2)-3/2*x^2*cosh(x)^2*polylog(2,-exp(2*x))*(a*sech(x)^4)^(1/2)+3/ 2*x^2*cosh(x)^2*polylog(2,exp(2*x))*(a*sech(x)^4)^(1/2)+3/2*x*cosh(x)^2*po lylog(3,-exp(2*x))*(a*sech(x)^4)^(1/2)-3/2*x*cosh(x)^2*polylog(3,exp(2*x)) *(a*sech(x)^4)^(1/2)-3/4*cosh(x)^2*polylog(4,-exp(2*x))*(a*sech(x)^4)^(1/2 )+3/4*cosh(x)^2*polylog(4,exp(2*x))*(a*sech(x)^4)^(1/2)-3/2*x^2*cosh(x)*si nh(x)*(a*sech(x)^4)^(1/2)-1/2*x^3*sinh(x)^2*(a*sech(x)^4)^(1/2)
Time = 0.62 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.46 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\frac {1}{4} \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (6 x^2+4 x^3 \log \left (1-e^{-2 x}\right )+12 x \log \left (1+e^{-2 x}\right )-4 x^3 \log \left (1+e^{-2 x}\right )+6 \left (-1+x^2\right ) \operatorname {PolyLog}\left (2,-e^{-2 x}\right )-6 x^2 \operatorname {PolyLog}\left (2,e^{-2 x}\right )+6 x \operatorname {PolyLog}\left (3,-e^{-2 x}\right )-6 x \operatorname {PolyLog}\left (3,e^{-2 x}\right )+3 \operatorname {PolyLog}\left (4,-e^{-2 x}\right )-3 \operatorname {PolyLog}\left (4,e^{-2 x}\right )+2 x^3 \text {sech}^2(x)-6 x^2 \tanh (x)\right ) \]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(6*x^2 + 4*x^3*Log[1 - E^(-2*x)] + 12*x*Log[1 + E^(-2*x)] - 4*x^3*Log[1 + E^(-2*x)] + 6*(-1 + x^2)*PolyLog[2, -E^(-2*x) ] - 6*x^2*PolyLog[2, E^(-2*x)] + 6*x*PolyLog[3, -E^(-2*x)] - 6*x*PolyLog[3 , E^(-2*x)] + 3*PolyLog[4, -E^(-2*x)] - 3*PolyLog[4, E^(-2*x)] + 2*x^3*Sec h[x]^2 - 6*x^2*Tanh[x]))/4
Time = 0.91 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.52, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {7271, 5985, 27, 2010, 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.
| \(\displaystyle \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int x^3 \text {csch}(x) \text {sech}^3(x)dx\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-3 \int \frac {1}{2} x^2 \left (2 \log (\tanh (x))-\tanh ^2(x)\right )dx-\frac {1}{2} x^3 \tanh ^2(x)+x^3 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-\frac {3}{2} \int x^2 \left (2 \log (\tanh (x))-\tanh ^2(x)\right )dx-\frac {1}{2} x^3 \tanh ^2(x)+x^3 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-\frac {3}{2} \int \left (2 x^2 \log (\tanh (x))-x^2 \tanh ^2(x)\right )dx-\frac {1}{2} x^3 \tanh ^2(x)+x^3 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-\frac {3}{2} \left (\frac {4}{3} x^3 \text {arctanh}\left (e^{2 x}\right )+x^2 \operatorname {PolyLog}\left (2,-e^{2 x}\right )-x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )-x \operatorname {PolyLog}\left (3,-e^{2 x}\right )+x \operatorname {PolyLog}\left (3,e^{2 x}\right )-\operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (4,-e^{2 x}\right )-\frac {\operatorname {PolyLog}\left (4,e^{2 x}\right )}{2}-\frac {x^3}{3}+\frac {2}{3} x^3 \log (\tanh (x))+x^2+x^2 \tanh (x)-2 x \log \left (e^{2 x}+1\right )\right )-\frac {1}{2} x^3 \tanh ^2(x)+x^3 \log (\tanh (x))\right )\) |
Cosh[x]^2*Sqrt[a*Sech[x]^4]*(x^3*Log[Tanh[x]] - (x^3*Tanh[x]^2)/2 - (3*(x^ 2 - x^3/3 + (4*x^3*ArcTanh[E^(2*x)])/3 - 2*x*Log[1 + E^(2*x)] + (2*x^3*Log [Tanh[x]])/3 - PolyLog[2, -E^(2*x)] + x^2*PolyLog[2, -E^(2*x)] - x^2*PolyL og[2, E^(2*x)] - x*PolyLog[3, -E^(2*x)] + x*PolyLog[3, E^(2*x)] + PolyLog[ 4, -E^(2*x)]/2 - PolyLog[4, E^(2*x)]/2 + x^2*Tanh[x]))/2)
3.9.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Simp[(c + d*x)^m u, x] - Simp[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]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(269)=538\).
Time = 0.12 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.85
| method | result | size |
| risch | \(\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} x^{2} \left (2 x \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x}+3\right )-3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{2}+3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \ln \left (1+{\mathrm e}^{2 x}\right )+\frac {3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 x}\right )}{2}+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{3} \ln \left (1+{\mathrm e}^{x}\right )+3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )-6 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )+6 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )-\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{3} \ln \left (1+{\mathrm e}^{2 x}\right )-\frac {3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 x}\right )}{2}+\frac {3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \operatorname {polylog}\left (3, -{\mathrm e}^{2 x}\right )}{2}-\frac {3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \operatorname {polylog}\left (4, -{\mathrm e}^{2 x}\right )}{4}+\sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{3} \ln \left (1-{\mathrm e}^{x}\right )+3 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-6 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+6 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )^{2} \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )\) | \(602\) |
(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*x^2*(2*x*exp(2*x)+3*exp(2*x)+3 )-3*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^2+3*(a*ex p(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x*ln(1+exp(2*x))+3/2 *(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*polylog(2,-exp (2*x))+(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^3*ln(1 +exp(x))+3*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^2* polylog(2,-exp(x))-6*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2* x))^2*x*polylog(3,-exp(x))+6*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*( 1+exp(2*x))^2*polylog(4,-exp(x))-(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2* x)*(1+exp(2*x))^2*x^3*ln(1+exp(2*x))-3/2*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2) *exp(-2*x)*(1+exp(2*x))^2*x^2*polylog(2,-exp(2*x))+3/2*(a*exp(4*x)/(1+exp( 2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x*polylog(3,-exp(2*x))-3/4*(a*exp( 4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*polylog(4,-exp(2*x))+( a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^3*ln(1-exp(x)) +3*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^2*polylog( 2,exp(x))-6*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x*p olylog(3,exp(x))+6*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x) )^2*polylog(4,exp(x))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 4629, normalized size of antiderivative = 14.20 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\text {Too large to display} \]
(6*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (c osh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2 *x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x ))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(4, cosh(x) + sinh(x)) - 6*((e^(4*x) + 2 *e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2* x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*( 3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cos h(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x) ^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cos h(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1 ))*e^(2*x)*polylog(4, I*cosh(x) + I*sinh(x)) - 6*((e^(4*x) + 2*e^(2*x) + 1 )*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x) )*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)* e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x )^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x ) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x...
\[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int x^{3} \sqrt {a \operatorname {sech}^{4}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.63 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=-3 \, \sqrt {a} x^{2} - \frac {1}{3} \, {\left (4 \, x^{3} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 6 \, x^{2} {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 6 \, x {\rm Li}_{3}(-e^{\left (2 \, x\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, x\right )})\right )} \sqrt {a} + {\left (x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})\right )} \sqrt {a} + {\left (x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})\right )} \sqrt {a} + \frac {3}{2} \, {\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} \sqrt {a} + \frac {3 \, \sqrt {a} x^{2} + {\left (2 \, \sqrt {a} x^{3} + 3 \, \sqrt {a} x^{2}\right )} e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \]
-3*sqrt(a)*x^2 - 1/3*(4*x^3*log(e^(2*x) + 1) + 6*x^2*dilog(-e^(2*x)) - 6*x *polylog(3, -e^(2*x)) + 3*polylog(4, -e^(2*x)))*sqrt(a) + (x^3*log(e^x + 1 ) + 3*x^2*dilog(-e^x) - 6*x*polylog(3, -e^x) + 6*polylog(4, -e^x))*sqrt(a) + (x^3*log(-e^x + 1) + 3*x^2*dilog(e^x) - 6*x*polylog(3, e^x) + 6*polylog (4, e^x))*sqrt(a) + 3/2*(2*x*log(e^(2*x) + 1) + dilog(-e^(2*x)))*sqrt(a) + (3*sqrt(a)*x^2 + (2*sqrt(a)*x^3 + 3*sqrt(a)*x^2)*e^(2*x))/(e^(4*x) + 2*e^ (2*x) + 1)
\[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{4}} x^{3} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
Timed out. \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int \frac {x^3\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]