3.43 \(\int \frac{\text{PolyLog}(3,a x^2)}{x^4} \, dx\)

Optimal. Leaf size=70 \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^2\right )}{3 x^3}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{8 a}{27 x} \]

[Out]

(-8*a)/(27*x) + (8*a^(3/2)*ArcTanh[Sqrt[a]*x])/27 + (4*Log[1 - a*x^2])/(27*x^3) - (2*PolyLog[2, a*x^2])/(9*x^3
) - PolyLog[3, a*x^2]/(3*x^3)

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Rubi [A]  time = 0.0409967, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2455, 325, 206} \[ -\frac{2 \text{PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^2\right )}{3 x^3}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{8 a}{27 x} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^2]/x^4,x]

[Out]

(-8*a)/(27*x) + (8*a^(3/2)*ArcTanh[Sqrt[a]*x])/27 + (4*Log[1 - a*x^2])/(27*x^3) - (2*PolyLog[2, a*x^2])/(9*x^3
) - PolyLog[3, a*x^2]/(3*x^3)

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^4} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{2}{3} \int \frac{\text{Li}_2\left (a x^2\right )}{x^4} \, dx\\ &=-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}-\frac{4}{9} \int \frac{\log \left (1-a x^2\right )}{x^4} \, dx\\ &=\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{1}{27} (8 a) \int \frac{1}{x^2 \left (1-a x^2\right )} \, dx\\ &=-\frac{8 a}{27 x}+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}+\frac{1}{27} \left (8 a^2\right ) \int \frac{1}{1-a x^2} \, dx\\ &=-\frac{8 a}{27 x}+\frac{8}{27} a^{3/2} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{4 \log \left (1-a x^2\right )}{27 x^3}-\frac{2 \text{Li}_2\left (a x^2\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^2\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.088813, size = 61, normalized size = 0.87 \[ -\frac{6 \text{PolyLog}\left (2,a x^2\right )+9 \text{PolyLog}\left (3,a x^2\right )-8 a^{3/2} x^3 \tanh ^{-1}\left (\sqrt{a} x\right )+8 a x^2-4 \log \left (1-a x^2\right )}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x^2]/x^4,x]

[Out]

-(8*a*x^2 - 8*a^(3/2)*x^3*ArcTanh[Sqrt[a]*x] - 4*Log[1 - a*x^2] + 6*PolyLog[2, a*x^2] + 9*PolyLog[3, a*x^2])/(
27*x^3)

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Maple [B]  time = 0.177, size = 125, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2}}{2} \left ( -{\frac{16}{27\,x}{\frac{1}{\sqrt{-a}}}}-{\frac{8\,ax}{27} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{a{x}^{2}}}}}+{\frac{8\,\ln \left ( -a{x}^{2}+1 \right ) }{27\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}}-{\frac{4\,{\it polylog} \left ( 2,a{x}^{2} \right ) }{9\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}}-{\frac{2\,{\it polylog} \left ( 3,a{x}^{2} \right ) }{3\,{x}^{3}a}{\frac{1}{\sqrt{-a}}}} \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^4,x)

[Out]

-1/2*a^2/(-a)^(1/2)*(-16/27/x/(-a)^(1/2)-8/27*x/(-a)^(1/2)*a/(a*x^2)^(1/2)*(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(
1/2)))+8/27/x^3/(-a)^(1/2)/a*ln(-a*x^2+1)-4/9/x^3/(-a)^(1/2)*polylog(2,a*x^2)/a-2/3/x^3/(-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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.22669, size = 462, normalized size = 6.6 \begin{align*} \left [\frac{4 \, a^{\frac{3}{2}} x^{3} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) - 8 \, a x^{2} - 6 \,{\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 ) - 9 \,{\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}, -\frac{8 \, \sqrt{-a} a x^{3} \arctan \left (\sqrt{-a} x\right ) + 8 \, a x^{2} + 6 \,{\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 ) + 9 \,{\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^4,x, algorithm="fricas")

[Out]

[1/27*(4*a^(3/2)*x^3*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)) - 8*a*x^2 - 6*\%iint(a, x, -log(-a*x^2 + 1)/a,
-2*log(-a*x^2 + 1)/x) + 4*log(-a*x^2 + 1) - 9*polylog(3, a*x^2))/x^3, -1/27*(8*sqrt(-a)*a*x^3*arctan(sqrt(-a)*
x) + 8*a*x^2 + 6*\%iint(a, x, -log(-a*x^2 + 1)/a, -2*log(-a*x^2 + 1)/x) - 4*log(-a*x^2 + 1) + 9*polylog(3, a*x^
2))/x^3]

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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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**2)/x**4,x)

[Out]

Integral(polylog(3, a*x**2)/x**4, 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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^4,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^2)/x^4, x)