3.1.50 \(\int \frac {\text {PolyLog}(2,a x^q)}{x^3} \, dx\) [50]

Optimal. Leaf size=78 \[ -\frac {a q^2 x^{-2+q} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}+\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\text {PolyLog}\left (2,a x^q\right )}{2 x^2} \]

[Out]

-1/4*a*q^2*x^(-2+q)*hypergeom([1, (-2+q)/q],[2-2/q],a*x^q)/(2-q)+1/4*q*ln(1-a*x^q)/x^2-1/2*polylog(2,a*x^q)/x^
2

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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6726, 2505, 371} \begin {gather*} -\frac {a q^2 x^{q-2} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}+\frac {q \log \left (1-a x^q\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q]/x^3,x]

[Out]

-1/4*(a*q^2*x^(-2 + q)*Hypergeometric2F1[1, -((2 - q)/q), 2*(1 - q^(-1)), a*x^q])/(2 - q) + (q*Log[1 - a*x^q])
/(4*x^2) - PolyLog[2, a*x^q]/(2*x^2)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -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\left (a x^q\right )}{x^3} \, dx &=-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}-\frac {1}{2} q \int \frac {\log \left (1-a x^q\right )}{x^3} \, dx\\ &=\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}+\frac {1}{4} \left (a q^2\right ) \int \frac {x^{-3+q}}{1-a x^q} \, dx\\ &=-\frac {a q^2 x^{-2+q} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}+\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.78 \begin {gather*} \frac {q \left (\frac {a q x^q \, _2F_1\left (1,\frac {-2+q}{q};2-\frac {2}{q};a x^q\right )}{-2+q}+\log \left (1-a x^q\right )\right )-2 \text {PolyLog}\left (2,a x^q\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^q]/x^3,x]

[Out]

(q*((a*q*x^q*Hypergeometric2F1[1, (-2 + q)/q, 2 - 2/q, a*x^q])/(-2 + q) + Log[1 - a*x^q]) - 2*PolyLog[2, a*x^q
])/(4*x^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 5.
time = 0.11, size = 108, normalized size = 1.38

method	result	size
meijerg	\(-\frac {\left (-a \right )^{\frac {2}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {2}{q}} \ln \left (1-a \,x^{q}\right )}{4 x^{2}}-\frac {q \left (-a \right )^{-\frac {2}{q}} \left (1-\frac {q}{2}\right ) \polylog \left (2, a \,x^{q}\right )}{\left (-2+q \right ) x^{2}}-\frac {q^{2} x^{-2+q} a \left (-a \right )^{-\frac {2}{q}} \Phi \left (a \,x^{q}, 1, \frac {-2+q}{q}\right )}{4}\right )}{q}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q)/x^3,x,method=_RETURNVERBOSE)

[Out]

-(-a)^(2/q)/q*(-1/4*q^2/x^2*(-a)^(-2/q)*ln(1-a*x^q)-q/(-2+q)/x^2*(-a)^(-2/q)*(1-1/2*q)*polylog(2,a*x^q)-1/4*q^
2*x^(-2+q)*a*(-a)^(-2/q)*LerchPhi(a*x^q,1,(-2+q)/q))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-q^2*integrate(1/4/(a*x^3*x^q - x^3), x) + 1/8*(q^2 + 2*q*log(-a*x^q + 1) - 4*dilog(a*x^q))/x^2

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Fricas [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^q)/x^3,x, algorithm="fricas")

[Out]

integral(dilog(a*x^q)/x^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(polylog(2, a*x**q)/x**3, x)

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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^q)/x^3,x, algorithm="giac")

[Out]

integrate(dilog(a*x^q)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x^q)/x^3,x)

[Out]

int(polylog(2, a*x^q)/x^3, x)

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