3.27 \(\int x^2 \text{PolyLog}(2,a x^2) \, dx\)

Optimal. Leaf size=63 \[ \frac{1}{3} x^3 \text{PolyLog}\left (2,a x^2\right )+\frac{4 \tanh ^{-1}\left (\sqrt{a} x\right )}{9 a^{3/2}}+\frac{2}{9} x^3 \log \left (1-a x^2\right )-\frac{4 x}{9 a}-\frac{4 x^3}{27} \]

[Out]

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

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Rubi [A]  time = 0.0387699, antiderivative size = 63, 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, 302, 206} \[ \frac{1}{3} x^3 \text{PolyLog}\left (2,a x^2\right )+\frac{4 \tanh ^{-1}\left (\sqrt{a} x\right )}{9 a^{3/2}}+\frac{2}{9} x^3 \log \left (1-a x^2\right )-\frac{4 x}{9 a}-\frac{4 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Int[x^2*PolyLog[2, a*x^2],x]

[Out]

(-4*x)/(9*a) - (4*x^3)/27 + (4*ArcTanh[Sqrt[a]*x])/(9*a^(3/2)) + (2*x^3*Log[1 - a*x^2])/9 + (x^3*PolyLog[2, a*
x^2])/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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 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 x^2 \text{Li}_2\left (a x^2\right ) \, dx &=\frac{1}{3} x^3 \text{Li}_2\left (a x^2\right )+\frac{2}{3} \int x^2 \log \left (1-a x^2\right ) \, dx\\ &=\frac{2}{9} x^3 \log \left (1-a x^2\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^2\right )+\frac{1}{9} (4 a) \int \frac{x^4}{1-a x^2} \, dx\\ &=\frac{2}{9} x^3 \log \left (1-a x^2\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^2\right )+\frac{1}{9} (4 a) \int \left (-\frac{1}{a^2}-\frac{x^2}{a}+\frac{1}{a^2 \left (1-a x^2\right )}\right ) \, dx\\ &=-\frac{4 x}{9 a}-\frac{4 x^3}{27}+\frac{2}{9} x^3 \log \left (1-a x^2\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^2\right )+\frac{4 \int \frac{1}{1-a x^2} \, dx}{9 a}\\ &=-\frac{4 x}{9 a}-\frac{4 x^3}{27}+\frac{4 \tanh ^{-1}\left (\sqrt{a} x\right )}{9 a^{3/2}}+\frac{2}{9} x^3 \log \left (1-a x^2\right )+\frac{1}{3} x^3 \text{Li}_2\left (a x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0514052, size = 57, normalized size = 0.9 \[ \frac{1}{27} \left (9 x^3 \text{PolyLog}\left (2,a x^2\right )+\frac{12 \tanh ^{-1}\left (\sqrt{a} x\right )}{a^{3/2}}+6 x^3 \log \left (1-a x^2\right )-\frac{12 x}{a}-4 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*PolyLog[2, a*x^2],x]

[Out]

((-12*x)/a - 4*x^3 + (12*ArcTanh[Sqrt[a]*x])/a^(3/2) + 6*x^3*Log[1 - a*x^2] + 9*x^3*PolyLog[2, a*x^2])/27

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Maple [A]  time = 0.046, size = 50, normalized size = 0.8 \begin{align*} -{\frac{4\,x}{9\,a}}-{\frac{4\,{x}^{3}}{27}}+{\frac{4}{9}{\it Artanh} \left ( x\sqrt{a} \right ){a}^{-{\frac{3}{2}}}}+{\frac{2\,{x}^{3}\ln \left ( -a{x}^{2}+1 \right ) }{9}}+{\frac{{x}^{3}{\it polylog} \left ( 2,a{x}^{2} \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*polylog(2,a*x^2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.63943, size = 351, normalized size = 5.57 \begin{align*} \left [\frac{9 \, a^{2} x^{3}{\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x + 6 \, \sqrt{a} \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right )}{27 \, a^{2}}, \frac{9 \, a^{2} x^{3}{\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x - 12 \, \sqrt{-a} \arctan \left (\sqrt{-a} x\right )}{27 \, a^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/27*(9*a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12*a*x + 6*sqrt(a)*log((a*x^2 + 2*sqrt
(a)*x + 1)/(a*x^2 - 1)))/a^2, 1/27*(9*a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12*a*x -
12*sqrt(-a)*arctan(sqrt(-a)*x))/a^2]

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Sympy [A]  time = 119.376, size = 83, normalized size = 1.32 \begin{align*} \begin{cases} - \frac{2 x^{3} \operatorname{Li}_{1}\left (a x^{2}\right )}{9} + \frac{x^{3} \operatorname{Li}_{2}\left (a x^{2}\right )}{3} - \frac{4 x^{3}}{27} - \frac{4 x}{9 a} - \frac{4 \log{\left (x - \sqrt{\frac{1}{a}} \right )}}{9 a^{2} \sqrt{\frac{1}{a}}} - \frac{2 \operatorname{Li}_{1}\left (a x^{2}\right )}{9 a^{2} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*x**3*polylog(1, a*x**2)/9 + x**3*polylog(2, a*x**2)/3 - 4*x**3/27 - 4*x/(9*a) - 4*log(x - sqrt(1
/a))/(9*a**2*sqrt(1/a)) - 2*polylog(1, a*x**2)/(9*a**2*sqrt(1/a)), Ne(a, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Li}_2\left (a x^{2}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*dilog(a*x^2), x)