Integrand size = 18, antiderivative size = 204 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\frac {1}{2} x^2 \cosh ^2(x) \sqrt {a \text {sech}^4(x)}-2 x^2 \text {arctanh}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt {a \text {sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt {a \text {sech}^4(x)}-x \cosh ^2(x) \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+x \cosh ^2(x) \operatorname {PolyLog}\left (2,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}+\frac {1}{2} \cosh ^2(x) \operatorname {PolyLog}\left (3,-e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-\frac {1}{2} \cosh ^2(x) \operatorname {PolyLog}\left (3,e^{2 x}\right ) \sqrt {a \text {sech}^4(x)}-x \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-\frac {1}{2} x^2 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \]
1/2*x^2*cosh(x)^2*(a*sech(x)^4)^(1/2)-2*x^2*arctanh(exp(2*x))*cosh(x)^2*(a *sech(x)^4)^(1/2)+cosh(x)^2*ln(cosh(x))*(a*sech(x)^4)^(1/2)-x*cosh(x)^2*po lylog(2,-exp(2*x))*(a*sech(x)^4)^(1/2)+x*cosh(x)^2*polylog(2,exp(2*x))*(a* sech(x)^4)^(1/2)+1/2*cosh(x)^2*polylog(3,-exp(2*x))*(a*sech(x)^4)^(1/2)-1/ 2*cosh(x)^2*polylog(3,exp(2*x))*(a*sech(x)^4)^(1/2)-x*cosh(x)*sinh(x)*(a*s ech(x)^4)^(1/2)-1/2*x^2*sinh(x)^2*(a*sech(x)^4)^(1/2)
Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\frac {1}{2} \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-2 x+2 x^2 \log \left (1-e^{-2 x}\right )-2 x^2 \log \left (1+e^{-2 x}\right )+2 \log \left (1+e^{2 x}\right )+2 x \operatorname {PolyLog}\left (2,-e^{-2 x}\right )-2 x \operatorname {PolyLog}\left (2,e^{-2 x}\right )+\operatorname {PolyLog}\left (3,-e^{-2 x}\right )-\operatorname {PolyLog}\left (3,e^{-2 x}\right )+x^2 \text {sech}^2(x)-2 x \tanh (x)\right ) \]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(-2*x + 2*x^2*Log[1 - E^(-2*x)] - 2*x^2*Log[1 + E^(-2*x)] + 2*Log[1 + E^(2*x)] + 2*x*PolyLog[2, -E^(-2*x)] - 2*x*PolyLo g[2, E^(-2*x)] + PolyLog[3, -E^(-2*x)] - PolyLog[3, E^(-2*x)] + x^2*Sech[x ]^2 - 2*x*Tanh[x]))/2
Time = 0.87 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.48, 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.
| \(\displaystyle \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int x^2 \text {csch}(x) \text {sech}^3(x)dx\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-2 \int \frac {1}{2} x \left (2 \log (\tanh (x))-\tanh ^2(x)\right )dx-\frac {1}{2} x^2 \tanh ^2(x)+x^2 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-\int x \left (2 \log (\tanh (x))-\tanh ^2(x)\right )dx-\frac {1}{2} x^2 \tanh ^2(x)+x^2 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-\int \left (2 x \log (\tanh (x))-x \tanh ^2(x)\right )dx-\frac {1}{2} x^2 \tanh ^2(x)+x^2 \log (\tanh (x))\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \left (-2 x^2 \text {arctanh}\left (e^{2 x}\right )-x \operatorname {PolyLog}\left (2,-e^{2 x}\right )+x \operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 x}\right )-\frac {\operatorname {PolyLog}\left (3,e^{2 x}\right )}{2}+\frac {x^2}{2}-\frac {1}{2} x^2 \tanh ^2(x)-x \tanh (x)+\log (\cosh (x))\right )\) |
Cosh[x]^2*Sqrt[a*Sech[x]^4]*(x^2/2 - 2*x^2*ArcTanh[E^(2*x)] + Log[Cosh[x]] - x*PolyLog[2, -E^(2*x)] + x*PolyLog[2, E^(2*x)] + PolyLog[3, -E^(2*x)]/2 - PolyLog[3, E^(2*x)]/2 - x*Tanh[x] - (x^2*Tanh[x]^2)/2)
3.9.52.3.1 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. \(440\) vs. \(2(173)=346\).
Time = 0.13 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.16
| method | result | size |
| risch | \(2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} x \left (x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x}+1\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} \ln \left ({\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} \ln \left (1+{\mathrm e}^{2 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^{2} \ln \left (1+{\mathrm e}^{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 \operatorname {polylog}\left (2, -{\mathrm e}^{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} \operatorname {polylog}\left (3, -{\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^{2} \ln \left (1+{\mathrm e}^{2 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 \operatorname {polylog}\left (2, -{\mathrm e}^{2 x}\right )+\frac {\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 (3, -{\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^{2} \ln \left (1-{\mathrm e}^{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 \operatorname {polylog}\left (2, {\mathrm e}^{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} \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )\) | \(441\) |
2*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*x*(x*exp(2*x)+exp(2*x)+1)-2* (a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*ln(exp(x))+(a*e xp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*ln(1+exp(2*x))+(a*e xp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^2*ln(1+exp(x))+2* (a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x*polylog(2,-ex p(x))-2*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*polylog (3,-exp(x))-(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2*x^2 *ln(1+exp(2*x))-(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1+exp(2*x))^2 *x*polylog(2,-exp(2*x))+1/2*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2*x)*(1 +exp(2*x))^2*polylog(3,-exp(2*x))+(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*exp(-2 *x)*(1+exp(2*x))^2*x^2*ln(1-exp(x))+2*(a*exp(4*x)/(1+exp(2*x))^4)^(1/2)*ex p(-2*x)*(1+exp(2*x))^2*x*polylog(2,exp(x))-2*(a*exp(4*x)/(1+exp(2*x))^4)^( 1/2)*exp(-2*x)*(1+exp(2*x))^2*polylog(3,exp(x))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 3431, normalized size of antiderivative = 16.82 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\text {Too large to display} \]
-(2*((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)*polylog(3, cosh(x) + sinh(x)) - 2*((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*co sh(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) + co sh(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(3, I*cosh(x) + I*sinh(x)) - 2*((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*...
\[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int x^{2} \sqrt {a \operatorname {sech}^{4}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.75 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=-\frac {1}{2} \, {\left (2 \, x^{2} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, x {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, x\right )})\right )} \sqrt {a} + {\left (x^{2} \log \left (e^{x} + 1\right ) + 2 \, x {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_{3}(-e^{x})\right )} \sqrt {a} + {\left (x^{2} \log \left (-e^{x} + 1\right ) + 2 \, x {\rm Li}_2\left (e^{x}\right ) - 2 \, {\rm Li}_{3}(e^{x})\right )} \sqrt {a} - 2 \, \sqrt {a} x + \sqrt {a} \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {2 \, {\left ({\left (\sqrt {a} x^{2} + \sqrt {a} x\right )} e^{\left (2 \, x\right )} + \sqrt {a} x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \]
-1/2*(2*x^2*log(e^(2*x) + 1) + 2*x*dilog(-e^(2*x)) - polylog(3, -e^(2*x))) *sqrt(a) + (x^2*log(e^x + 1) + 2*x*dilog(-e^x) - 2*polylog(3, -e^x))*sqrt( a) + (x^2*log(-e^x + 1) + 2*x*dilog(e^x) - 2*polylog(3, e^x))*sqrt(a) - 2* sqrt(a)*x + sqrt(a)*log(e^(2*x) + 1) + 2*((sqrt(a)*x^2 + sqrt(a)*x)*e^(2*x ) + sqrt(a)*x)/(e^(4*x) + 2*e^(2*x) + 1)
\[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{4}} x^{2} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
Timed out. \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^4(x)} \, dx=\int \frac {x^2\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]