3.25 \(\int \frac{\text{PolyLog}(2,a x^2)}{x^7} \, dx\)

Optimal. Leaf size=74 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{6 x^6}-\frac{a^2}{18 x^2}-\frac{1}{18} a^3 \log \left (1-a x^2\right )+\frac{1}{9} a^3 \log (x)-\frac{a}{36 x^4}+\frac{\log \left (1-a x^2\right )}{18 x^6} \]

[Out]

-a/(36*x^4) - a^2/(18*x^2) + (a^3*Log[x])/9 - (a^3*Log[1 - a*x^2])/18 + Log[1 - a*x^2]/(18*x^6) - PolyLog[2, a
*x^2]/(6*x^6)

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Rubi [A]  time = 0.0541201, antiderivative size = 74, 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, 2454, 2395, 44} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{6 x^6}-\frac{a^2}{18 x^2}-\frac{1}{18} a^3 \log \left (1-a x^2\right )+\frac{1}{9} a^3 \log (x)-\frac{a}{36 x^4}+\frac{\log \left (1-a x^2\right )}{18 x^6} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^2]/x^7,x]

[Out]

-a/(36*x^4) - a^2/(18*x^2) + (a^3*Log[x])/9 - (a^3*Log[1 - a*x^2])/18 + Log[1 - a*x^2]/(18*x^6) - PolyLog[2, a
*x^2]/(6*x^6)

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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\text{Li}_2\left (a x^2\right )}{x^7} \, dx &=-\frac{\text{Li}_2\left (a x^2\right )}{6 x^6}-\frac{1}{3} \int \frac{\log \left (1-a x^2\right )}{x^7} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{6 x^6}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{\log (1-a x)}{x^4} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{18 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{6 x^6}+\frac{1}{18} a \operatorname{Subst}\left (\int \frac{1}{x^3 (1-a x)} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{18 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{6 x^6}+\frac{1}{18} a \operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{a}{x^2}+\frac{a^2}{x}-\frac{a^3}{-1+a x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a}{36 x^4}-\frac{a^2}{18 x^2}+\frac{1}{9} a^3 \log (x)-\frac{1}{18} a^3 \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{18 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0314649, size = 60, normalized size = 0.81 \[ -\frac{6 \text{PolyLog}\left (2,a x^2\right )-4 a^3 x^6 \log (x)+2 \left (a^3 x^6-1\right ) \log \left (1-a x^2\right )+a x^2 \left (2 a x^2+1\right )}{36 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^2]/x^7,x]

[Out]

-(a*x^2*(1 + 2*a*x^2) - 4*a^3*x^6*Log[x] + 2*(-1 + a^3*x^6)*Log[1 - a*x^2] + 6*PolyLog[2, a*x^2])/(36*x^6)

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Maple [A]  time = 0.05, size = 62, normalized size = 0.8 \begin{align*} -{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{6\,{x}^{6}}}+{\frac{\ln \left ( -a{x}^{2}+1 \right ) }{18\,{x}^{6}}}-{\frac{a}{36\,{x}^{4}}}-{\frac{{a}^{2}}{18\,{x}^{2}}}+{\frac{{a}^{3}\ln \left ( x \right ) }{9}}-{\frac{{a}^{3}\ln \left ( a{x}^{2}-1 \right ) }{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^2)/x^7,x)

[Out]

-1/6*polylog(2,a*x^2)/x^6+1/18*ln(-a*x^2+1)/x^6-1/36*a/x^4-1/18*a^2/x^2+1/9*a^3*ln(x)-1/18*a^3*ln(a*x^2-1)

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Maxima [A]  time = 0.981148, size = 74, normalized size = 1. \begin{align*} \frac{1}{9} \, a^{3} \log \left (x\right ) - \frac{2 \, a^{2} x^{4} + a x^{2} + 2 \,{\left (a^{3} x^{6} - 1\right )} \log \left (-a x^{2} + 1\right ) + 6 \,{\rm Li}_2\left (a x^{2}\right )}{36 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^2)/x^7,x, algorithm="maxima")

[Out]

1/9*a^3*log(x) - 1/36*(2*a^2*x^4 + a*x^2 + 2*(a^3*x^6 - 1)*log(-a*x^2 + 1) + 6*dilog(a*x^2))/x^6

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Fricas [A]  time = 2.65693, size = 154, normalized size = 2.08 \begin{align*} -\frac{2 \, a^{3} x^{6} \log \left (a x^{2} - 1\right ) - 4 \, a^{3} x^{6} \log \left (x\right ) + 2 \, a^{2} x^{4} + a x^{2} + 6 \,{\rm Li}_2\left (a x^{2}\right ) - 2 \, \log \left (-a x^{2} + 1\right )}{36 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^2)/x^7,x, algorithm="fricas")

[Out]

-1/36*(2*a^3*x^6*log(a*x^2 - 1) - 4*a^3*x^6*log(x) + 2*a^2*x^4 + a*x^2 + 6*dilog(a*x^2) - 2*log(-a*x^2 + 1))/x
^6

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Sympy [A]  time = 43.0619, size = 58, normalized size = 0.78 \begin{align*} \frac{a^{3} \log{\left (x \right )}}{9} + \frac{a^{3} \operatorname{Li}_{1}\left (a x^{2}\right )}{18} - \frac{a^{2}}{18 x^{2}} - \frac{a}{36 x^{4}} - \frac{\operatorname{Li}_{1}\left (a x^{2}\right )}{18 x^{6}} - \frac{\operatorname{Li}_{2}\left (a x^{2}\right )}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x**2)/x**7,x)

[Out]

a**3*log(x)/9 + a**3*polylog(1, a*x**2)/18 - a**2/(18*x**2) - a/(36*x**4) - polylog(1, a*x**2)/(18*x**6) - pol
ylog(2, a*x**2)/(6*x**6)

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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^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x^2)/x^7,x, algorithm="giac")

[Out]

integrate(dilog(a*x^2)/x^7, x)