3.38 \(\int \frac{\text{PolyLog}(3,a x^2)}{x^7} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{18 x^6}-\frac{\text{PolyLog}\left (3,a x^2\right )}{6 x^6}-\frac{a^2}{54 x^2}-\frac{1}{54} a^3 \log \left (1-a x^2\right )+\frac{1}{27} a^3 \log (x)-\frac{a}{108 x^4}+\frac{\log \left (1-a x^2\right )}{54 x^6} \]

[Out]

-a/(108*x^4) - a^2/(54*x^2) + (a^3*Log[x])/27 - (a^3*Log[1 - a*x^2])/54 + Log[1 - a*x^2]/(54*x^6) - PolyLog[2,
a*x^2]/(18*x^6) - PolyLog[3, a*x^2]/(6*x^6)

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Rubi [A]  time = 0.0654848, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6591, 2454, 2395, 44} \[ -\frac{\text{PolyLog}\left (2,a x^2\right )}{18 x^6}-\frac{\text{PolyLog}\left (3,a x^2\right )}{6 x^6}-\frac{a^2}{54 x^2}-\frac{1}{54} a^3 \log \left (1-a x^2\right )+\frac{1}{27} a^3 \log (x)-\frac{a}{108 x^4}+\frac{\log \left (1-a x^2\right )}{54 x^6} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x^2]/x^7,x]

[Out]

-a/(108*x^4) - a^2/(54*x^2) + (a^3*Log[x])/27 - (a^3*Log[1 - a*x^2])/54 + Log[1 - a*x^2]/(54*x^6) - PolyLog[2,
a*x^2]/(18*x^6) - PolyLog[3, 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}_3\left (a x^2\right )}{x^7} \, dx &=-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}+\frac{1}{3} \int \frac{\text{Li}_2\left (a x^2\right )}{x^7} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{18 x^6}-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}-\frac{1}{9} \int \frac{\log \left (1-a x^2\right )}{x^7} \, dx\\ &=-\frac{\text{Li}_2\left (a x^2\right )}{18 x^6}-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}-\frac{1}{18} \operatorname{Subst}\left (\int \frac{\log (1-a x)}{x^4} \, dx,x,x^2\right )\\ &=\frac{\log \left (1-a x^2\right )}{54 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{18 x^6}-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}+\frac{1}{54} 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 )}{54 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{18 x^6}-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}+\frac{1}{54} 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}{108 x^4}-\frac{a^2}{54 x^2}+\frac{1}{27} a^3 \log (x)-\frac{1}{54} a^3 \log \left (1-a x^2\right )+\frac{\log \left (1-a x^2\right )}{54 x^6}-\frac{\text{Li}_2\left (a x^2\right )}{18 x^6}-\frac{\text{Li}_3\left (a x^2\right )}{6 x^6}\\ \end{align*}

Mathematica [C]  time = 0.0119825, size = 30, normalized size = 0.34 \[ \frac{G_{5,5}^{2,4}\left (-a x^2|\begin{array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end{array}\right )}{2 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[PolyLog[3, a*x^2]/x^7,x]

[Out]

MeijerG[{{1, 1, 1, 1}, {4}}, {{1, 3}, {0, 0, 0}}, -(a*x^2)]/(2*x^6)

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Maple [A]  time = 0.066, size = 115, normalized size = 1.3 \begin{align*}{\frac{{a}^{3}}{2} \left ( -{\frac{1}{2\,{a}^{2}{x}^{4}}}-{\frac{1}{8\,a{x}^{2}}}-{\frac{1}{27}}+{\frac{2\,\ln \left ( x \right ) }{27}}+{\frac{\ln \left ( -a \right ) }{27}}+{\frac{64\,{a}^{2}{x}^{4}+152\,a{x}^{2}+832}{1728\,{a}^{2}{x}^{4}}}+{\frac{ \left ( -64\,{x}^{6}{a}^{3}+64 \right ) \ln \left ( -a{x}^{2}+1 \right ) }{1728\,{x}^{6}{a}^{3}}}-{\frac{{\it polylog} \left ( 2,a{x}^{2} \right ) }{9\,{x}^{6}{a}^{3}}}-{\frac{{\it polylog} \left ( 3,a{x}^{2} \right ) }{3\,{x}^{6}{a}^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x^2)/x^7,x)

[Out]

1/2*a^3*(-1/2/a^2/x^4-1/8/a/x^2-1/27+2/27*ln(x)+1/27*ln(-a)+1/1728/a^2/x^4*(64*a^2*x^4+152*a*x^2+832)+1/1728/a
^3/x^6*(-64*a^3*x^6+64)*ln(-a*x^2+1)-1/9/a^3/x^6*polylog(2,a*x^2)-1/3/a^3/x^6*polylog(3,a*x^2))

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Maxima [A]  time = 0.997547, size = 86, normalized size = 0.98 \begin{align*} \frac{1}{27} \, 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 ) + 18 \,{\rm Li}_{3}(a x^{2})}{108 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^7,x, algorithm="maxima")

[Out]

1/27*a^3*log(x) - 1/108*(2*a^2*x^4 + a*x^2 + 2*(a^3*x^6 - 1)*log(-a*x^2 + 1) + 6*dilog(a*x^2) + 18*polylog(3,
a*x^2))/x^6

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Fricas [C]  time = 2.79682, size = 242, normalized size = 2.75 \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 \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) - 2 \, \log \left (-a x^{2} + 1\right ) + 18 \,{\rm polylog}\left (3, a x^{2}\right )}{108 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^7,x, algorithm="fricas")

[Out]

-1/108*(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*\%iint(a, x, -log(-a*x^2 + 1)/a, -2
*log(-a*x^2 + 1)/x) - 2*log(-a*x^2 + 1) + 18*polylog(3, a*x^2))/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x^{2}\right )}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x**2)/x**7,x)

[Out]

Integral(polylog(3, a*x**2)/x**7, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x^{2})}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x^2)/x^7,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x^2)/x^7, x)