3.1.29 \(\int \frac {\text {PolyLog}(2,a x^2)}{x^2} \, dx\) [29]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {6726, 2505, 212} \begin {gather*} -\frac {\text {Li}_2\left (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 ) \end {gather*}

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 212

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

Rule 2505

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 6726

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]

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*}

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.98 \begin {gather*} \frac {4 \sqrt {a} x \tanh ^{-1}\left (\sqrt {a} x\right )+2 \log \left (1-a x^2\right )-\text {PolyLog}\left (2,a x^2\right )}{x} \end {gather*}

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 = 0.07, size = 39, normalized size = 0.93

method	result	size
default	\(\frac {2 \ln \left (-a \,x^{2}+1\right )}{x}-\frac {\polylog \left (2, a \,x^{2}\right )}{x}+4 \arctanh \left (x \sqrt {a}\right ) \sqrt {a}\)	\(39\)
meijerg	\(\frac {a \left (-\frac {4 x \sqrt {-a}\, \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{\sqrt {a \,x^{2}}}+\frac {4 \sqrt {-a}\, \ln \left (-a \,x^{2}+1\right )}{x a}-\frac {2 \sqrt {-a}\, \polylog \left (2, a \,x^{2}\right )}{x a}\right )}{2 \sqrt {-a}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^2)/x^2,x,method=_RETURNVERBOSE)

[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 [A]
time = 0.46, size = 49, normalized size = 1.17 \begin {gather*} -2 \, \sqrt {a} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(a)*log((a*x - sqrt(a))/(a*x + sqrt(a))) - (dilog(a*x^2) - 2*log(-a*x^2 + 1))/x

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Fricas [A]
time = 0.60, size = 94, normalized size = 2.24 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (34) = 68\).
time = 12.26, size = 184, normalized size = 4.38 \begin {gather*} \begin {cases} - \frac {\pi ^{2}}{6 x} & \text {for}\: a = \frac {1}{x^{2}} \\0 & \text {for}\: a = 0 \\- \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-pi**2/(6*x), Eq(a, x**(-2))), (0, Eq(a, 0)), (-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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.26, size = 38, normalized size = 0.90 \begin {gather*} 4\,\sqrt {a}\,\mathrm {atanh}\left (\sqrt {a}\,x\right )-\frac {\mathrm {polylog}\left (2,a\,x^2\right )}{x}+\frac {2\,\ln \left (1-a\,x^2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x^2)/x^2,x)

[Out]

4*a^(1/2)*atanh(a^(1/2)*x) - polylog(2, a*x^2)/x + (2*log(1 - a*x^2))/x

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