Integrand size = 18, antiderivative size = 287 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=x^3 \sqrt {a \text {sech}^2(x)}-6 x^2 \arctan \left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^3 \text {arctanh}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-3 x^2 \cosh (x) \operatorname {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+6 i x \cosh (x) \operatorname {PolyLog}\left (2,-i e^x\right ) \sqrt {a \text {sech}^2(x)}-6 i x \cosh (x) \operatorname {PolyLog}\left (2,i e^x\right ) \sqrt {a \text {sech}^2(x)}+3 x^2 \cosh (x) \operatorname {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)}+6 x \cosh (x) \operatorname {PolyLog}\left (3,-e^x\right ) \sqrt {a \text {sech}^2(x)}-6 i \cosh (x) \operatorname {PolyLog}\left (3,-i e^x\right ) \sqrt {a \text {sech}^2(x)}+6 i \cosh (x) \operatorname {PolyLog}\left (3,i e^x\right ) \sqrt {a \text {sech}^2(x)}-6 x \cosh (x) \operatorname {PolyLog}\left (3,e^x\right ) \sqrt {a \text {sech}^2(x)}-6 \cosh (x) \operatorname {PolyLog}\left (4,-e^x\right ) \sqrt {a \text {sech}^2(x)}+6 \cosh (x) \operatorname {PolyLog}\left (4,e^x\right ) \sqrt {a \text {sech}^2(x)} \]
x^3*(a*sech(x)^2)^(1/2)-6*x^2*arctan(exp(x))*cosh(x)*(a*sech(x)^2)^(1/2)-2 *x^3*arctanh(exp(x))*cosh(x)*(a*sech(x)^2)^(1/2)-3*x^2*cosh(x)*polylog(2,- exp(x))*(a*sech(x)^2)^(1/2)+6*I*x*cosh(x)*polylog(2,-I*exp(x))*(a*sech(x)^ 2)^(1/2)-6*I*x*cosh(x)*polylog(2,I*exp(x))*(a*sech(x)^2)^(1/2)+3*x^2*cosh( x)*polylog(2,exp(x))*(a*sech(x)^2)^(1/2)+6*x*cosh(x)*polylog(3,-exp(x))*(a *sech(x)^2)^(1/2)-6*I*cosh(x)*polylog(3,-I*exp(x))*(a*sech(x)^2)^(1/2)+6*I *cosh(x)*polylog(3,I*exp(x))*(a*sech(x)^2)^(1/2)-6*x*cosh(x)*polylog(3,exp (x))*(a*sech(x)^2)^(1/2)-6*cosh(x)*polylog(4,-exp(x))*(a*sech(x)^2)^(1/2)+ 6*cosh(x)*polylog(4,exp(x))*(a*sech(x)^2)^(1/2)
Time = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.63 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\left (x^3-3 i \cosh (x) \left (x^2 \log \left (1-i e^x\right )-x^2 \log \left (1+i e^x\right )-2 x \operatorname {PolyLog}\left (2,-i e^x\right )+2 x \operatorname {PolyLog}\left (2,i e^x\right )+2 \operatorname {PolyLog}\left (3,-i e^x\right )-2 \operatorname {PolyLog}\left (3,i e^x\right )\right )+\cosh (x) \left (x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right )\right ) \sqrt {a \text {sech}^2(x)} \]
(x^3 - (3*I)*Cosh[x]*(x^2*Log[1 - I*E^x] - x^2*Log[1 + I*E^x] - 2*x*PolyLo g[2, (-I)*E^x] + 2*x*PolyLog[2, I*E^x] + 2*PolyLog[3, (-I)*E^x] - 2*PolyLo g[3, I*E^x]) + Cosh[x]*(x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLo g[2, -E^x] + 3*x^2*PolyLog[2, E^x] + 6*x*PolyLog[3, -E^x] - 6*x*PolyLog[3, E^x] - 6*PolyLog[4, -E^x] + 6*PolyLog[4, E^x]))*Sqrt[a*Sech[x]^2]
Time = 0.82 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.59, 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.
| \(\displaystyle \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \int x^3 \text {csch}(x) \text {sech}^2(x)dx\) |
| \(\Big \downarrow \) 5985 |
| \(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (-3 \int -x^2 (\text {arctanh}(\cosh (x))-\text {sech}(x))dx+x^3 (-\text {arctanh}(\cosh (x)))+x^3 \text {sech}(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (3 \int x^2 (\text {arctanh}(\cosh (x))-\text {sech}(x))dx+x^3 (-\text {arctanh}(\cosh (x)))+x^3 \text {sech}(x)\right )\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (3 \int \left (x^2 \text {arctanh}(\cosh (x))-x^2 \text {sech}(x)\right )dx+x^3 (-\text {arctanh}(\cosh (x)))+x^3 \text {sech}(x)\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \cosh (x) \sqrt {a \text {sech}^2(x)} \left (3 \left (-2 x^2 \arctan \left (e^x\right )-\frac {2}{3} x^3 \text {arctanh}\left (e^x\right )+\frac {1}{3} x^3 \text {arctanh}(\cosh (x))-x^2 \operatorname {PolyLog}\left (2,-e^x\right )+x^2 \operatorname {PolyLog}\left (2,e^x\right )+2 i x \operatorname {PolyLog}\left (2,-i e^x\right )-2 i x \operatorname {PolyLog}\left (2,i e^x\right )+2 x \operatorname {PolyLog}\left (3,-e^x\right )-2 x \operatorname {PolyLog}\left (3,e^x\right )-2 i \operatorname {PolyLog}\left (3,-i e^x\right )+2 i \operatorname {PolyLog}\left (3,i e^x\right )-2 \operatorname {PolyLog}\left (4,-e^x\right )+2 \operatorname {PolyLog}\left (4,e^x\right )\right )+x^3 (-\text {arctanh}(\cosh (x)))+x^3 \text {sech}(x)\right )\) |
Cosh[x]*Sqrt[a*Sech[x]^2]*(-(x^3*ArcTanh[Cosh[x]]) + 3*(-2*x^2*ArcTan[E^x] - (2*x^3*ArcTanh[E^x])/3 + (x^3*ArcTanh[Cosh[x]])/3 - x^2*PolyLog[2, -E^x ] + (2*I)*x*PolyLog[2, (-I)*E^x] - (2*I)*x*PolyLog[2, I*E^x] + x^2*PolyLog [2, E^x] + 2*x*PolyLog[3, -E^x] - (2*I)*PolyLog[3, (-I)*E^x] + (2*I)*PolyL og[3, I*E^x] - 2*x*PolyLog[3, E^x] - 2*PolyLog[4, -E^x] + 2*PolyLog[4, E^x ]) + x^3*Sech[x])
3.9.50.3.1 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])
\[\int x^{3} \operatorname {csch}\left (x \right ) \operatorname {sech}\left (x \right ) \sqrt {a \operatorname {sech}\left (x \right )^{2}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (229) = 458\).
Time = 0.29 (sec) , antiderivative size = 1202, normalized size of antiderivative = 4.19 \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\text {Too large to display} \]
(6*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cos h(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x *polylog(4, cosh(x) + sinh(x)) - 6*((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(4, -cosh(x) - sinh(x)) - 6*(x*cos h(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)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^ x*polylog(3, cosh(x) + sinh(x)) - 6*((-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)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, I*cosh(x) + I*si nh(x)) - 6*((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)*sqrt(a/(e^(4*x) + 2 *e^(2*x) + 1))*e^x*polylog(3, -I*cosh(x) - I*sinh(x)) + 6*(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)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog( 3, -cosh(x) - sinh(x)) + (2*x^3*cosh(x)*e^(2*x) + 2*x^3*cosh(x) + 3*(x^2*c osh(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))*dilog(cosh(x) + sinh (x)) - 6*(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)*di...
\[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int x^{3} \sqrt {a \operatorname {sech}^{2}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}\, dx \]
\[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{2}} x^{3} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
2*sqrt(a)*x^3*e^x/(e^(2*x) + 1) - (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) - 12*s qrt(a)*integrate(1/2*x^2*e^x/(e^(2*x) + 1), x)
\[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int { \sqrt {a \operatorname {sech}\left (x\right )^{2}} x^{3} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right ) \,d x } \]
Timed out. \[ \int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx=\int \frac {x^3\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )} \,d x \]