3.1.9 \(\int \frac {\text {PolyLog}(2,a x)}{x^4} \, dx\) [9]

Optimal. Leaf size=68 \[ -\frac {a}{18 x^2}-\frac {a^2}{9 x}+\frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)+\frac {\log (1-a x)}{9 x^3}-\frac {\text {PolyLog}(2,a x)}{3 x^3} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6726, 2442, 46} \begin {gather*} \frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)-\frac {a^2}{9 x}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {\log (1-a x)}{9 x^3}-\frac {a}{18 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x]/x^4,x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2442

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 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(a x)}{x^4} \, dx &=-\frac {\text {Li}_2(a x)}{3 x^3}-\frac {1}{3} \int \frac {\log (1-a x)}{x^4} \, dx\\ &=\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {1}{9} a \int \frac {1}{x^3 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}+\frac {1}{9} 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}{18 x^2}-\frac {a^2}{9 x}+\frac {1}{9} a^3 \log (x)-\frac {1}{9} a^3 \log (1-a x)+\frac {\log (1-a x)}{9 x^3}-\frac {\text {Li}_2(a x)}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 52, normalized size = 0.76 \begin {gather*} -\frac {a x (1+2 a x)-2 a^3 x^3 \log (x)+2 \left (-1+a^3 x^3\right ) \log (1-a x)+6 \text {PolyLog}(2,a x)}{18 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x]/x^4,x]

[Out]

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

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Maple [A]
time = 0.45, size = 76, normalized size = 1.12

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\polylog \left (2, a x \right )}{3 a^{3} x^{3}}-\frac {1}{9 a x}-\frac {1}{18 a^{2} x^{2}}+\frac {\ln \left (-a x \right )}{9}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right ) \left (\left (-a x +1\right )^{2}+3 a x \right )}{9 a^{3} x^{3}}\right )\)	\(76\)
default	\(a^{3} \left (-\frac {\polylog \left (2, a x \right )}{3 a^{3} x^{3}}-\frac {1}{9 a x}-\frac {1}{18 a^{2} x^{2}}+\frac {\ln \left (-a x \right )}{9}+\frac {\ln \left (-a x +1\right ) \left (-a x +1\right ) \left (\left (-a x +1\right )^{2}+3 a x \right )}{9 a^{3} x^{3}}\right )\)	\(76\)
meijerg	\(a^{3} \left (\frac {32 a^{2} x^{2}+60 a x +192}{432 a^{2} x^{2}}+\frac {\left (-16 a^{3} x^{3}+16\right ) \ln \left (-a x +1\right )}{144 a^{3} x^{3}}-\frac {\polylog \left (2, a x \right )}{3 a^{3} x^{3}}-\frac {2}{27}+\frac {\ln \left (x \right )}{9}+\frac {\ln \left (-a \right )}{9}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{4 a x}\right )\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 49, normalized size = 0.72 \begin {gather*} \frac {1}{9} \, 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 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x^4,x, algorithm="maxima")

[Out]

1/9*a^3*log(x) - 1/18*(2*a^2*x^2 + a*x + 2*(a^3*x^3 - 1)*log(-a*x + 1) + 6*dilog(a*x))/x^3

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Fricas [A]
time = 0.39, size = 56, normalized size = 0.82 \begin {gather*} -\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 Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )}{18 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.23, size = 51, normalized size = 0.75 \begin {gather*} \frac {a^{3} \log {\left (x \right )}}{9} + \frac {a^{3} \operatorname {Li}_{1}\left (a x\right )}{9} - \frac {a^{2}}{9 x} - \frac {a}{18 x^{2}} - \frac {\operatorname {Li}_{1}\left (a x\right )}{9 x^{3}} - \frac {\operatorname {Li}_{2}\left (a x\right )}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x)/x**4,x)

[Out]

a**3*log(x)/9 + a**3*polylog(1, a*x)/9 - a**2/(9*x) - a/(18*x**2) - polylog(1, a*x)/(9*x**3) - polylog(2, a*x)
/(3*x**3)

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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)/x^4,x, algorithm="giac")

[Out]

integrate(dilog(a*x)/x^4, x)

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Mupad [B]
time = 0.32, size = 57, normalized size = 0.84 \begin {gather*} \frac {2\,a^3\,\ln \left (x\right )}{9}-\frac {\frac {a\,x}{18}-\frac {\ln \left (1-a\,x\right )}{9}+\frac {\mathrm {polylog}\left (2,a\,x\right )}{3}+\frac {a^2\,x^2}{9}}{x^3}-\frac {a^3\,\ln \left (a\,x^2-x\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x)/x^4,x)

[Out]

(2*a^3*log(x))/9 - ((a*x)/18 - log(1 - a*x)/9 + polylog(2, a*x)/3 + (a^2*x^2)/9)/x^3 - (a^3*log(a*x^2 - x))/9

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