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

Optimal. Leaf size=42 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{x}+\frac{2 \log \left (1-a x^2\right )}{x}+4 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0261791, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 206} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{x}+\frac{2 \log \left (1-a x^2\right )}{x}+4 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^2]/x^2,x]

[Out]

4*Sqrt[a]*ArcTanh[Sqrt[a]*x] + (2*Log[1 - a*x^2])/x - PolyLog[2, 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}_2\left (a x^2\right )}{x^2} \, dx &=-\frac{\text{Li}_2\left (a x^2\right )}{x}-2 \int \frac{\log \left (1-a x^2\right )}{x^2} \, dx\\ &=\frac{2 \log \left (1-a x^2\right )}{x}-\frac{\text{Li}_2\left (a x^2\right )}{x}+(4 a) \int \frac{1}{1-a x^2} \, dx\\ &=4 \sqrt{a} \tanh ^{-1}\left (\sqrt{a} x\right )+\frac{2 \log \left (1-a x^2\right )}{x}-\frac{\text{Li}_2\left (a x^2\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.0177403, size = 41, normalized size = 0.98 \[ \frac{-\text{PolyLog}\left (2,a x^2\right )+2 \log \left (1-a x^2\right )+4 \sqrt{a} x \tanh ^{-1}\left (\sqrt{a} x\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^2]/x^2,x]

[Out]

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

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Maple [A]  time = 0.049, size = 39, normalized size = 0.9 \begin{align*} 2\,{\frac{\ln \left ( -a{x}^{2}+1 \right ) }{x}}-{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{x}}+4\,{\it Artanh} \left ( x\sqrt{a} \right ) \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(2,a*x^2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.79204, size = 232, normalized size = 5.52 \begin{align*} \left [\frac{2 \, \sqrt{a} x \log \left (\frac{a x^{2} + 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right ) -{\rm Li}_2\left (a x^{2}\right ) + 2 \, \log \left (-a x^{2} + 1\right )}{x}, -\frac{4 \, \sqrt{-a} x \arctan \left (\sqrt{-a} x\right ) +{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(2*sqrt(a)*x*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)) - dilog(a*x^2) + 2*log(-a*x^2 + 1))/x, -(4*sqrt(-a)*x
*arctan(sqrt(-a)*x) + dilog(a*x^2) - 2*log(-a*x^2 + 1))/x]

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Sympy [A]  time = 84.144, size = 184, normalized size = 4.38 \begin{align*} \begin{cases} 0 & \text{for}\: a = 0 \\- \frac{\pi ^{2}}{6 x} & \text{for}\: a = \frac{1}{x^{2}} \\- \frac{4 a x^{3} \sqrt{\frac{1}{a}} \log{\left (x - \sqrt{\frac{1}{a}} \right )}}{x^{3} - \frac{x}{a}} - \frac{2 a x^{3} \sqrt{\frac{1}{a}} \operatorname{Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac{x}{a}} - \frac{2 x^{2} \operatorname{Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac{x}{a}} - \frac{x^{2} \operatorname{Li}_{2}\left (a x^{2}\right )}{x^{3} - \frac{x}{a}} + \frac{4 x \sqrt{\frac{1}{a}} \log{\left (x - \sqrt{\frac{1}{a}} \right )}}{x^{3} - \frac{x}{a}} + \frac{2 x \sqrt{\frac{1}{a}} \operatorname{Li}_{1}\left (a x^{2}\right )}{x^{3} - \frac{x}{a}} + \frac{2 \operatorname{Li}_{1}\left (a x^{2}\right )}{a x^{3} - x} + \frac{\operatorname{Li}_{2}\left (a x^{2}\right )}{a x^{3} - x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((0, Eq(a, 0)), (-pi**2/(6*x), Eq(a, x**(-2))), (-4*a*x**3*sqrt(1/a)*log(x - sqrt(1/a))/(x**3 - x/a)
- 2*a*x**3*sqrt(1/a)*polylog(1, a*x**2)/(x**3 - x/a) - 2*x**2*polylog(1, a*x**2)/(x**3 - x/a) - x**2*polylog(2
, a*x**2)/(x**3 - x/a) + 4*x*sqrt(1/a)*log(x - sqrt(1/a))/(x**3 - x/a) + 2*x*sqrt(1/a)*polylog(1, a*x**2)/(x**
3 - x/a) + 2*polylog(1, a*x**2)/(a*x**3 - x) + polylog(2, a*x**2)/(a*x**3 - x), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x^2)/x^2, x)