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

Optimal. Leaf size=36 \[ -\frac{\text{PolyLog}(2,a x)}{x}+a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x} \]

[Out]

a*Log[x] - a*Log[1 - a*x] + Log[1 - a*x]/x - PolyLog[2, a*x]/x

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Rubi [A]  time = 0.0218984, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {6591, 2395, 36, 29, 31} \[ -\frac{\text{PolyLog}(2,a x)}{x}+a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/x^2,x]

[Out]

a*Log[x] - a*Log[1 - a*x] + Log[1 - a*x]/x - PolyLog[2, a*x]/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 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\text{Li}_2(a x)}{x^2} \, dx &=-\frac{\text{Li}_2(a x)}{x}-\int \frac{\log (1-a x)}{x^2} \, dx\\ &=\frac{\log (1-a x)}{x}-\frac{\text{Li}_2(a x)}{x}+a \int \frac{1}{x (1-a x)} \, dx\\ &=\frac{\log (1-a x)}{x}-\frac{\text{Li}_2(a x)}{x}+a \int \frac{1}{x} \, dx+a^2 \int \frac{1}{1-a x} \, dx\\ &=a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x}-\frac{\text{Li}_2(a x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.0097221, size = 36, normalized size = 1. \[ -\frac{\text{PolyLog}(2,a x)}{x}+a \log (x)-a \log (1-a x)+\frac{\log (1-a x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/x^2,x]

[Out]

a*Log[x] - a*Log[1 - a*x] + Log[1 - a*x]/x - PolyLog[2, a*x]/x

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Maple [A]  time = 0.119, size = 40, normalized size = 1.1 \begin{align*} -{\frac{{\it polylog} \left ( 2,ax \right ) }{x}}+a\ln \left ( -ax \right ) -a\ln \left ( -ax+1 \right ) +{\frac{\ln \left ( -ax+1 \right ) }{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00914, size = 38, normalized size = 1.06 \begin{align*} a \log \left (x\right ) - \frac{{\left (a x - 1\right )} \log \left (-a x + 1\right ) +{\rm Li}_2\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - ((a*x - 1)*log(-a*x + 1) + dilog(a*x))/x

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Fricas [A]  time = 2.78236, size = 88, normalized size = 2.44 \begin{align*} -\frac{a x \log \left (a x - 1\right ) - a x \log \left (x\right ) +{\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*x*log(a*x - 1) - a*x*log(x) + dilog(a*x) - log(-a*x + 1))/x

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Sympy [A]  time = 1.56346, size = 24, normalized size = 0.67 \begin{align*} a \log{\left (x \right )} + a \operatorname{Li}_{1}\left (a x\right ) - \frac{\operatorname{Li}_{1}\left (a x\right )}{x} - \frac{\operatorname{Li}_{2}\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x)/x^2, x)