Integrand size = 18, antiderivative size = 187 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=x^2 \sqrt {a \text {sech}^2(x)}-4 x \arctan \left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \text {arctanh}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \operatorname {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+2 i \cosh (x) \operatorname {PolyLog}\left (2,-i e^x\right ) \sqrt {a \text {sech}^2(x)}-2 i \cosh (x) \operatorname {PolyLog}\left (2,i e^x\right ) \sqrt {a \text {sech}^2(x)}+2 x \cosh (x) \operatorname {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)}+2 \cosh (x) \operatorname {PolyLog}\left (3,-e^x\right ) \sqrt {a \text {sech}^2(x)}-2 \cosh (x) \operatorname {PolyLog}\left (3,e^x\right ) \sqrt {a \text {sech}^2(x)} \]
x^2*(a*sech(x)^2)^(1/2)-4*x*arctan(exp(x))*cosh(x)*(a*sech(x)^2)^(1/2)-2*x ^2*arctanh(exp(x))*cosh(x)*(a*sech(x)^2)^(1/2)-2*x*cosh(x)*polylog(2,-exp( x))*(a*sech(x)^2)^(1/2)+2*I*cosh(x)*polylog(2,-I*exp(x))*(a*sech(x)^2)^(1/ 2)-2*I*cosh(x)*polylog(2,I*exp(x))*(a*sech(x)^2)^(1/2)+2*x*cosh(x)*polylog (2,exp(x))*(a*sech(x)^2)^(1/2)+2*cosh(x)*polylog(3,-exp(x))*(a*sech(x)^2)^ (1/2)-2*cosh(x)*polylog(3,exp(x))*(a*sech(x)^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\left (x^2-2 i \cosh (x) \left (x \left (\log \left (1-i e^x\right )-\log \left (1+i e^x\right )\right )-\operatorname {PolyLog}\left (2,-i e^x\right )+\operatorname {PolyLog}\left (2,i e^x\right )\right )+\cosh (x) \left (x^2 \log \left (1-e^x\right )-x^2 \log \left (1+e^x\right )-2 x \operatorname {PolyLog}\left (2,-e^x\right )+2 x \operatorname {PolyLog}\left (2,e^x\right )+2 \operatorname {PolyLog}\left (3,-e^x\right )-2 \operatorname {PolyLog}\left (3,e^x\right )\right )\right ) \sqrt {a \text {sech}^2(x)} \]
(x^2 - (2*I)*Cosh[x]*(x*(Log[1 - I*E^x] - Log[1 + I*E^x]) - PolyLog[2, (-I )*E^x] + PolyLog[2, I*E^x]) + Cosh[x]*(x^2*Log[1 - E^x] - x^2*Log[1 + E^x] - 2*x*PolyLog[2, -E^x] + 2*x*PolyLog[2, E^x] + 2*PolyLog[3, -E^x] - 2*Pol yLog[3, E^x]))*Sqrt[a*Sech[x]^2]
Time = 0.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.61, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {7271, 5985, 25, 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.
\(\displaystyle \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \int x^2 \text {csch}(x) \text {sech}^2(x)dx\) |
\(\Big \downarrow \) 5985 |
\(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (-2 \int -x (\text {arctanh}(\cosh (x))-\text {sech}(x))dx+x^2 (-\text {arctanh}(\cosh (x)))+x^2 \text {sech}(x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (2 \int x (\text {arctanh}(\cosh (x))-\text {sech}(x))dx+x^2 (-\text {arctanh}(\cosh (x)))+x^2 \text {sech}(x)\right )\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (2 \int (x \text {arctanh}(\cosh (x))-x \text {sech}(x))dx+x^2 (-\text {arctanh}(\cosh (x)))+x^2 \text {sech}(x)\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (2 \left (-2 x \arctan \left (e^x\right )+x^2 \left (-\text {arctanh}\left (e^x\right )\right )+\frac {1}{2} x^2 \text {arctanh}(\cosh (x))-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+i \operatorname {PolyLog}\left (2,-i e^x\right )-i \operatorname {PolyLog}\left (2,i e^x\right )+\operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (3,e^x\right )\right )+x^2 (-\text {arctanh}(\cosh (x)))+x^2 \text {sech}(x)\right )\) |
Cosh[x]*Sqrt[a*Sech[x]^2]*(-(x^2*ArcTanh[Cosh[x]]) + 2*(-2*x*ArcTan[E^x] - x^2*ArcTanh[E^x] + (x^2*ArcTanh[Cosh[x]])/2 - x*PolyLog[2, -E^x] + I*Poly Log[2, (-I)*E^x] - I*PolyLog[2, I*E^x] + x*PolyLog[2, E^x] + PolyLog[3, -E ^x] - PolyLog[3, E^x]) + x^2*Sech[x])
3.9.49.3.1 Defintions of rubi rules used
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])
\[\int x^{2} \operatorname {csch}\left (x \right ) \operatorname {sech}\left (x \right ) \sqrt {a \operatorname {sech}\left (x \right )^{2}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (149) = 298\).
Time = 0.28 (sec) , antiderivative size = 786, normalized size of antiderivative = 4.20 \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\text {Too large to display} \]
-(2*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(co sh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^ x*polylog(3, cosh(x) + sinh(x)) - 2*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt (a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, -cosh(x) - sinh(x)) - (2*x^2* cosh(x)*e^(2*x) + 2*x^2*cosh(x) + 2*(x*cosh(x)^2 + (x*e^(2*x) + x)*sinh(x) ^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x)*dilog(cosh(x) + sinh(x)) - 2*((I*e^(2*x) + I)*sinh(x)^2 + I*cosh(x)^ 2 + (I*cosh(x)^2 + I)*e^(2*x) + 2*(I*cosh(x)*e^(2*x) + I*cosh(x))*sinh(x) + I)*dilog(I*cosh(x) + I*sinh(x)) - 2*((-I*e^(2*x) - I)*sinh(x)^2 - I*cosh (x)^2 + (-I*cosh(x)^2 - I)*e^(2*x) + 2*(-I*cosh(x)*e^(2*x) - I*cosh(x))*si nh(x) - I)*dilog(-I*cosh(x) - I*sinh(x)) - 2*(x*cosh(x)^2 + (x*e^(2*x) + x )*sinh(x)^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x) )*sinh(x) + x)*dilog(-cosh(x) - sinh(x)) - (x^2*cosh(x)^2 + (x^2*e^(2*x) + x^2)*sinh(x)^2 + x^2 + (x^2*cosh(x)^2 + x^2)*e^(2*x) + 2*(x^2*cosh(x)*e^( 2*x) + x^2*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - 2*(-I*x*cosh(x)^ 2 + (-I*x*e^(2*x) - I*x)*sinh(x)^2 + (-I*x*cosh(x)^2 - I*x)*e^(2*x) + 2*(- I*x*cosh(x)*e^(2*x) - I*x*cosh(x))*sinh(x) - I*x)*log(I*cosh(x) + I*sinh(x ) + 1) - 2*(I*x*cosh(x)^2 + (I*x*e^(2*x) + I*x)*sinh(x)^2 + (I*x*cosh(x)^2 + I*x)*e^(2*x) + 2*(I*x*cosh(x)*e^(2*x) + I*x*cosh(x))*sinh(x) + I*x)*...
\[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int x^{2} \sqrt {a \operatorname {sech}^{2}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}\, dx \]
\[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{2}} x^{2} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
2*sqrt(a)*x^2*e^x/(e^(2*x) + 1) - (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*polylo g(3, e^x))*sqrt(a) - 4*sqrt(a)*integrate(x*e^x/(e^(2*x) + 1), x)
\[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{2}} x^{2} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
Timed out. \[ \int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int \frac {x^2\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]