3.44 \(\int \frac{\text{PolyLog}(3,a x^2)}{x^6} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{25 x^5}-\frac{\text{PolyLog}\left (3,a x^2\right )}{5 x^5}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5} \]

[Out]

(-8*a)/(375*x^3) - (8*a^2)/(125*x) + (8*a^(5/2)*ArcTanh[Sqrt[a]*x])/125 + (4*Log[1 - a*x^2])/(125*x^5) - (2*Po
lyLog[2, a*x^2])/(25*x^5) - PolyLog[3, a*x^2]/(5*x^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0469064, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2455, 325, 206} \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{25 x^5}-\frac{\text{PolyLog}\left (3,a x^2\right )}{5 x^5}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^2]/x^6,x]

[Out]

(-8*a)/(375*x^3) - (8*a^2)/(125*x) + (8*a^(5/2)*ArcTanh[Sqrt[a]*x])/125 + (4*Log[1 - a*x^2])/(125*x^5) - (2*Po
lyLog[2, a*x^2])/(25*x^5) - PolyLog[3, a*x^2]/(5*x^5)

Rule 6591

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[((d*x)^(m + 1)*PolyLog[n
, a*(b*x^p)^q])/(d*(m + 1)), x] - Dist[(p*q)/(m + 1), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ
[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\text{Li}_3\left (a x^2\right )}{x^6} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{2}{5} \int \frac{\text{Li}_2\left (a x^2\right )}{x^6} \, dx\\ &=-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}-\frac{4}{25} \int \frac{\log \left (1-a x^2\right )}{x^6} \, dx\\ &=\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} (8 a) \int \frac{1}{x^4 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{375 x^3}+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} \left (8 a^2\right ) \int \frac{1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{375 x^3}-\frac{8 a^2}{125 x}+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}+\frac{1}{125} \left (8 a^3\right ) \int \frac{1}{1-a x^2} \, dx\\ &=-\frac{8 a}{375 x^3}-\frac{8 a^2}{125 x}+\frac{8}{125} a^{5/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{125 x^5}-\frac{2 \text{Li}_2\left (a x^2\right )}{25 x^5}-\frac{\text{Li}_3\left (a x^2\right )}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.100693, size = 69, normalized size = 0.86 \[ -\frac{30 \text{PolyLog}\left (2,a x^2\right )+75 \text{PolyLog}\left (3,a x^2\right )+24 a^2 x^4-24 a^{5/2} x^5 \tanh ^{-1}\left (\sqrt{a} x\right )+8 a x^2-12 \log \left (1-a x^2\right )}{375 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^2]/x^6,x]

[Out]

-(8*a*x^2 + 24*a^2*x^4 - 24*a^(5/2)*x^5*ArcTanh[Sqrt[a]*x] - 12*Log[1 - a*x^2] + 30*PolyLog[2, a*x^2] + 75*Pol
yLog[3, a*x^2])/(375*x^5)

________________________________________________________________________________________

Maple [B]  time = 0.18, size = 138, normalized size = 1.7 \begin{align*}{\frac{{a}^{3}}{2} \left ( -{\frac{16}{375\,{x}^{3}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,a}{125\,x} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{a}^{2}x}{125} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ) \left ( -a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a{x}^{2}}}}}+{\frac{8\,\ln \left ( -a{x}^{2}+1 \right ) }{125\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{25\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{5\,a{x}^{5}} \left ( -a \right ) ^{-{\frac{3}{2}}}} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x^2)/x^6,x)

[Out]

1/2*a^3/(-a)^(1/2)*(-16/375/x^3/(-a)^(3/2)-16/125/x/(-a)^(3/2)*a-8/125*x/(-a)^(3/2)*a^2/(a*x^2)^(1/2)*(ln(1-(a
*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))+8/125/x^5/(-a)^(3/2)/a*ln(-a*x^2+1)-4/25/x^5/(-a)^(3/2)*polylog(2,a*x^2)/a-2
/5/x^5/(-a)^(3/2)/a*polylog(3,a*x^2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 3.24127, size = 513, normalized size = 6.41 \begin{align*} \left [\frac{12 \, a^{\frac{5}{2}} x^{5} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 24 \, a^{2} x^{4} - 8 \, a x^{2} - 30 \,{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) + 12 \, \log \left (-a x^{2} + 1\right ) - 75 \,{\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}, -\frac{24 \, \sqrt{-a} a^{2} x^{5} \arctan \left (\sqrt{-a} x\right ) + 24 \, a^{2} x^{4} + 8 \, a x^{2} + 30 \,{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) - 12 \, \log \left (-a x^{2} + 1\right ) + 75 \,{\rm polylog}\left (3, a x^{2}\right )}{375 \, x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^6,x, algorithm="fricas")

[Out]

[1/375*(12*a^(5/2)*x^5*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)) - 24*a^2*x^4 - 8*a*x^2 - 30*\%iint(a, x, -log
(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) + 12*log(-a*x^2 + 1) - 75*polylog(3, a*x^2))/x^5, -1/375*(24*sqrt(-a)*a^
2*x^5*arctan(sqrt(-a)*x) + 24*a^2*x^4 + 8*a*x^2 + 30*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) - 1
2*log(-a*x^2 + 1) + 75*polylog(3, a*x^2))/x^5]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x^{2}\right )}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**2)/x**6,x)

[Out]

Integral(polylog(3, a*x**2)/x**6, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x^{2})}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^6,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^2)/x^6, x)