Integrand size = 18, antiderivative size = 150 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {2 x^3 \text {arctanh}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 x \operatorname {PolyLog}\left (3,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 x \operatorname {PolyLog}\left (3,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 \operatorname {PolyLog}\left (4,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 \operatorname {PolyLog}\left (4,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \]
-2*x^3*arctanh(exp(x))*sech(x)/(a*sech(x)^2)^(1/2)-3*x^2*polylog(2,-exp(x) )*sech(x)/(a*sech(x)^2)^(1/2)+3*x^2*polylog(2,exp(x))*sech(x)/(a*sech(x)^2 )^(1/2)+6*x*polylog(3,-exp(x))*sech(x)/(a*sech(x)^2)^(1/2)-6*x*polylog(3,e xp(x))*sech(x)/(a*sech(x)^2)^(1/2)-6*polylog(4,-exp(x))*sech(x)/(a*sech(x) ^2)^(1/2)+6*polylog(4,exp(x))*sech(x)/(a*sech(x)^2)^(1/2)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\left (x^3 \log \left (1-e^x\right )-x^3 \log \left (1+e^x\right )-3 x^2 \operatorname {PolyLog}\left (2,-e^x\right )+3 x^2 \operatorname {PolyLog}\left (2,e^x\right )+6 x \operatorname {PolyLog}\left (3,-e^x\right )-6 x \operatorname {PolyLog}\left (3,e^x\right )-6 \operatorname {PolyLog}\left (4,-e^x\right )+6 \operatorname {PolyLog}\left (4,e^x\right )\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \]
((x^3*Log[1 - E^x] - x^3*Log[1 + E^x] - 3*x^2*PolyLog[2, -E^x] + 3*x^2*Pol yLog[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])*Sech[x])/Sqrt[a*Sech[x]^2]
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {7271, 3042, 26, 4670, 3011, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \frac {\text {sech}(x) \int x^3 \text {csch}(x)dx}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {sech}(x) \int i x^3 \csc (i x)dx}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \text {sech}(x) \int x^3 \csc (i x)dx}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle \frac {i \text {sech}(x) \left (3 i \int x^2 \log \left (1-e^x\right )dx-3 i \int x^2 \log \left (1+e^x\right )dx+2 i x^3 \text {arctanh}\left (e^x\right )\right )}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {i \text {sech}(x) \left (-3 i \left (2 \int x \operatorname {PolyLog}\left (2,-e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \int x \operatorname {PolyLog}\left (2,e^x\right )dx-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {i \text {sech}(x) \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int \operatorname {PolyLog}\left (3,-e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int \operatorname {PolyLog}\left (3,e^x\right )dx\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {i \text {sech}(x) \left (-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,-e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\int e^{-x} \operatorname {PolyLog}\left (3,e^x\right )de^x\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )+2 i x^3 \text {arctanh}\left (e^x\right )\right )}{\sqrt {a \text {sech}^2(x)}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {i \text {sech}(x) \left (2 i x^3 \text {arctanh}\left (e^x\right )-3 i \left (2 \left (x \operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (4,-e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,-e^x\right )\right )+3 i \left (2 \left (x \operatorname {PolyLog}\left (3,e^x\right )-\operatorname {PolyLog}\left (4,e^x\right )\right )-x^2 \operatorname {PolyLog}\left (2,e^x\right )\right )\right )}{\sqrt {a \text {sech}^2(x)}}\) |
(I*((2*I)*x^3*ArcTanh[E^x] - (3*I)*(-(x^2*PolyLog[2, -E^x]) + 2*(x*PolyLog [3, -E^x] - PolyLog[4, -E^x])) + (3*I)*(-(x^2*PolyLog[2, E^x]) + 2*(x*Poly Log[3, E^x] - PolyLog[4, E^x])))*Sech[x])/Sqrt[a*Sech[x]^2]
3.9.44.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
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] - Simp[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]
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. \(280\) vs. \(2(129)=258\).
Time = 0.09 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {{\mathrm e}^{x} x^{3} \ln \left (1+{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {3 \,{\mathrm e}^{x} x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {3 \,{\mathrm e}^{x} x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\) | \(281\) |
-1/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(1+exp(x))- 3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^2*polylog(2,-exp (x))+6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*polylog(3,- exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*polylog(4, -exp(x))+1/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(1- exp(x))+3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^2*polylo g(2,exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*poly log(3,exp(x))+6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*poly log(4,exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (127) = 254\).
Time = 0.26 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.81 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {{\left (6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 6 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 6 \, {\left (x e^{\left (2 \, x\right )} + x\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + {\left (3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}\right )} e^{\left (-x\right )}}{a} \]
(6*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog(4, cosh(x) + sinh(x)) - 6*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog (4, -cosh(x) - sinh(x)) - 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, cosh(x) + sinh(x)) + 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, -cosh(x) - sinh(x)) + (3*(x^2*e^(2*x) + x ^2)*dilog(cosh(x) + sinh(x)) - 3*(x^2*e^(2*x) + x^2)*dilog(-cosh(x) - sinh (x)) - (x^3*e^(2*x) + x^3)*log(cosh(x) + sinh(x) + 1) + (x^3*e^(2*x) + x^3 )*log(-cosh(x) - sinh(x) + 1))*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x)*e^(- x)/a
\[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\int \frac {x^{3} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\left (x \right )}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.53 \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) - 6 \, x {\rm Li}_{3}(-e^{x}) + 6 \, {\rm Li}_{4}(-e^{x})}{\sqrt {a}} + \frac {x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 6 \, x {\rm Li}_{3}(e^{x}) + 6 \, {\rm Li}_{4}(e^{x})}{\sqrt {a}} \]
-(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)
\[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\int { \frac {x^{3} \operatorname {csch}\left (x\right ) \operatorname {sech}\left (x\right )}{\sqrt {a \operatorname {sech}\left (x\right )^{2}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx=\int \frac {x^3}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \]