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

Optimal. Leaf size=54 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{x}-\frac{\text{PolyLog}\left (3,a x^2\right )}{x}+\frac{4 \log \left (1-a x^2\right )}{x}+8 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right ) \]

[Out]

8*Sqrt[a]*ArcTanh[Sqrt[a]*x] + (4*Log[1 - a*x^2])/x - (2*PolyLog[2, a*x^2])/x - PolyLog[3, a*x^2]/x

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Rubi [A]  time = 0.0357649, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 206} \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{x}-\frac{\text{PolyLog}\left (3,a x^2\right )}{x}+\frac{4 \log \left (1-a x^2\right )}{x}+8 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^2]/x^2,x]

[Out]

8*Sqrt[a]*ArcTanh[Sqrt[a]*x] + (4*Log[1 - a*x^2])/x - (2*PolyLog[2, a*x^2])/x - PolyLog[3, a*x^2]/x

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

Mathematica [A]  time = 0.0806658, size = 50, normalized size = 0.93 \[ \frac{-2 \text{PolyLog}\left (2,a x^2\right )-\text{PolyLog}\left (3,a x^2\right )+4 \log \left (1-a x^2\right )+8 \sqrt{a} x \tanh ^{-1}\left (\sqrt{a} x\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^2]/x^2,x]

[Out]

(8*Sqrt[a]*x*ArcTanh[Sqrt[a]*x] + 4*Log[1 - a*x^2] - 2*PolyLog[2, a*x^2] - PolyLog[3, a*x^2])/x

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Maple [B]  time = 0.172, size = 112, normalized size = 2.1 \begin{align*}{\frac{a}{2} \left ( -8\,{\frac{x\sqrt{-a} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ) }{\sqrt{a{x}^{2}}}}+8\,{\frac{\sqrt{-a}\ln \left ( -a{x}^{2}+1 \right ) }{ax}}-4\,{\frac{\sqrt{-a}{\it polylog} \left ( 2,a{x}^{2} \right ) }{ax}}-2\,{\frac{\sqrt{-a}{\it polylog} \left ( 3,a{x}^{2} \right ) }{ax}} \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^2,x)

[Out]

1/2*a/(-a)^(1/2)*(-8*x*(-a)^(1/2)/(a*x^2)^(1/2)*(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))+8/x*(-a)^(1/2)/a*ln(
-a*x^2+1)-4/x*(-a)^(1/2)*polylog(2,a*x^2)/a-2/x*(-a)^(1/2)/a*polylog(3,a*x^2))

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 2.87069, size = 402, normalized size = 7.44 \begin{align*} \left [\frac{4 \, \sqrt{a} x \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 2 \,{\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 ) + 4 \, \log \left (-a x^{2} + 1\right ) -{\rm polylog}\left (3, a x^{2}\right )}{x}, -\frac{8 \, \sqrt{-a} x \arctan \left (\sqrt{-a} x\right ) + 2 \,{\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 ) - 4 \, \log \left (-a x^{2} + 1\right ) +{\rm polylog}\left (3, a x^{2}\right )}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(3, a*x**2)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(polylog(3, a*x^2)/x^2, x)