3.18 \(\int \frac{\text{PolyLog}(3,a x)}{x^4} \, dx\)

Optimal. Leaf size=80 \[ -\frac{\text{PolyLog}(2,a x)}{9 x^3}-\frac{\text{PolyLog}(3,a x)}{3 x^3}-\frac{a^2}{27 x}+\frac{1}{27} a^3 \log (x)-\frac{1}{27} a^3 \log (1-a x)-\frac{a}{54 x^2}+\frac{\log (1-a x)}{27 x^3} \]

[Out]

-a/(54*x^2) - a^2/(27*x) + (a^3*Log[x])/27 - (a^3*Log[1 - a*x])/27 + Log[1 - a*x]/(27*x^3) - PolyLog[2, a*x]/(
9*x^3) - PolyLog[3, a*x]/(3*x^3)

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Rubi [A]  time = 0.047769, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6591, 2395, 44} \[ -\frac{\text{PolyLog}(2,a x)}{9 x^3}-\frac{\text{PolyLog}(3,a x)}{3 x^3}-\frac{a^2}{27 x}+\frac{1}{27} a^3 \log (x)-\frac{1}{27} a^3 \log (1-a x)-\frac{a}{54 x^2}+\frac{\log (1-a x)}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x]/x^4,x]

[Out]

-a/(54*x^2) - a^2/(27*x) + (a^3*Log[x])/27 - (a^3*Log[1 - a*x])/27 + Log[1 - a*x]/(27*x^3) - PolyLog[2, a*x]/(
9*x^3) - PolyLog[3, a*x]/(3*x^3)

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 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(a x)}{x^4} \, dx &=-\frac{\text{Li}_3(a x)}{3 x^3}+\frac{1}{3} \int \frac{\text{Li}_2(a x)}{x^4} \, dx\\ &=-\frac{\text{Li}_2(a x)}{9 x^3}-\frac{\text{Li}_3(a x)}{3 x^3}-\frac{1}{9} \int \frac{\log (1-a x)}{x^4} \, dx\\ &=\frac{\log (1-a x)}{27 x^3}-\frac{\text{Li}_2(a x)}{9 x^3}-\frac{\text{Li}_3(a x)}{3 x^3}+\frac{1}{27} a \int \frac{1}{x^3 (1-a x)} \, dx\\ &=\frac{\log (1-a x)}{27 x^3}-\frac{\text{Li}_2(a x)}{9 x^3}-\frac{\text{Li}_3(a x)}{3 x^3}+\frac{1}{27} a \int \left (\frac{1}{x^3}+\frac{a}{x^2}+\frac{a^2}{x}-\frac{a^3}{-1+a x}\right ) \, dx\\ &=-\frac{a}{54 x^2}-\frac{a^2}{27 x}+\frac{1}{27} a^3 \log (x)-\frac{1}{27} a^3 \log (1-a x)+\frac{\log (1-a x)}{27 x^3}-\frac{\text{Li}_2(a x)}{9 x^3}-\frac{\text{Li}_3(a x)}{3 x^3}\\ \end{align*}

Mathematica [C]  time = 0.00906, size = 25, normalized size = 0.31 \[ \frac{G_{5,5}^{2,4}\left (-a x\left |\begin{array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end{array}\right .\right )}{x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[PolyLog[3, a*x]/x^4,x]

[Out]

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

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Maple [A]  time = 0.168, size = 106, normalized size = 1.3 \begin{align*}{a}^{3} \left ( -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{1}{8\,ax}}-{\frac{1}{27}}+{\frac{\ln \left ( x \right ) }{27}}+{\frac{\ln \left ( -a \right ) }{27}}+{\frac{64\,{a}^{2}{x}^{2}+152\,ax+832}{1728\,{a}^{2}{x}^{2}}}+{\frac{ \left ( -64\,{x}^{3}{a}^{3}+64 \right ) \ln \left ( -ax+1 \right ) }{1728\,{x}^{3}{a}^{3}}}-{\frac{{\it polylog} \left ( 2,ax \right ) }{9\,{x}^{3}{a}^{3}}}-{\frac{{\it polylog} \left ( 3,ax \right ) }{3\,{x}^{3}{a}^{3}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.988239, size = 76, normalized size = 0.95 \begin{align*} \frac{1}{27} \, a^{3} \log \left (x\right ) - \frac{2 \, a^{2} x^{2} + a x + 2 \,{\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right ) + 6 \,{\rm Li}_2\left (a x\right ) + 18 \,{\rm Li}_{3}(a x)}{54 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/x^4,x, algorithm="maxima")

[Out]

1/27*a^3*log(x) - 1/54*(2*a^2*x^2 + a*x + 2*(a^3*x^3 - 1)*log(-a*x + 1) + 6*dilog(a*x) + 18*polylog(3, a*x))/x
^3

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Fricas [C]  time = 2.81394, size = 221, normalized size = 2.76 \begin{align*} -\frac{2 \, a^{3} x^{3} \log \left (a x - 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) + 2 \, a^{2} x^{2} + a x + 6 \,{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 2 \, \log \left (-a x + 1\right ) + 18 \,{\rm polylog}\left (3, a x\right )}{54 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/x^4,x, algorithm="fricas")

[Out]

-1/54*(2*a^3*x^3*log(a*x - 1) - 2*a^3*x^3*log(x) + 2*a^2*x^2 + a*x + 6*\%iint(a, x, -log(-a*x + 1)/a, -log(-a*x
+ 1)/x) - 2*log(-a*x + 1) + 18*polylog(3, a*x))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x\right )}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/x**4,x)

[Out]

Integral(polylog(3, a*x)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(a x)}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x)/x^4,x, algorithm="giac")

[Out]

integrate(polylog(3, a*x)/x^4, x)