3.51 \(\int \frac{\text{PolyLog}(2,a x^q)}{x^4} \, dx\)

Optimal. Leaf size=76 \[ -\frac{a q^2 x^{q-3} \text{Hypergeometric2F1}\left (1,-\frac{3-q}{q},2-\frac{3}{q},a x^q\right )}{9 (3-q)}-\frac{\text{PolyLog}\left (2,a x^q\right )}{3 x^3}+\frac{q \log \left (1-a x^q\right )}{9 x^3} \]

[Out]

-(a*q^2*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(9*(3 - q)) + (q*Log[1 - a*x^q])/(9*x^3
) - PolyLog[2, a*x^q]/(3*x^3)

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Rubi [A]  time = 0.040045, antiderivative size = 76, normalized size of antiderivative = 1., 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 = {6591, 2455, 364} \[ -\frac{\text{PolyLog}\left (2,a x^q\right )}{3 x^3}-\frac{a q^2 x^{q-3} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{9 (3-q)}+\frac{q \log \left (1-a x^q\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x^q]/x^4,x]

[Out]

-(a*q^2*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(9*(3 - q)) + (q*Log[1 - a*x^q])/(9*x^3
) - PolyLog[2, a*x^q]/(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 2455

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{\text{Li}_2\left (a x^q\right )}{x^4} \, dx &=-\frac{\text{Li}_2\left (a x^q\right )}{3 x^3}-\frac{1}{3} q \int \frac{\log \left (1-a x^q\right )}{x^4} \, dx\\ &=\frac{q \log \left (1-a x^q\right )}{9 x^3}-\frac{\text{Li}_2\left (a x^q\right )}{3 x^3}+\frac{1}{9} \left (a q^2\right ) \int \frac{x^{-4+q}}{1-a x^q} \, dx\\ &=-\frac{a q^2 x^{-3+q} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{9 (3-q)}+\frac{q \log \left (1-a x^q\right )}{9 x^3}-\frac{\text{Li}_2\left (a x^q\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0518445, size = 61, normalized size = 0.8 \[ \frac{q \left (\frac{a q x^q \text{Hypergeometric2F1}\left (1,\frac{q-3}{q},2-\frac{3}{q},a x^q\right )}{q-3}+\log \left (1-a x^q\right )\right )-3 \text{PolyLog}\left (2,a x^q\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x^q]/x^4,x]

[Out]

(q*((a*q*x^q*Hypergeometric2F1[1, (-3 + q)/q, 2 - 3/q, a*x^q])/(-3 + q) + Log[1 - a*x^q]) - 3*PolyLog[2, a*x^q
])/(9*x^3)

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Maple [C]  time = 0.205, size = 108, normalized size = 1.4 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{3\,{q}^{-1}} \left ( -{\frac{{q}^{2}\ln \left ( 1-a{x}^{q} \right ) }{9\,{x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}}}-{\frac{q{\it polylog} \left ( 2,a{x}^{q} \right ) }{ \left ( -3+q \right ){x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}} \left ( 1-{\frac{q}{3}} \right ) }-{\frac{{q}^{2}{x}^{-3+q}a}{9} \left ( -a \right ) ^{-3\,{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{-3+q}{q}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x^q)/x^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -q^{2} \int \frac{1}{9 \,{\left (a x^{4} x^{q} - x^{4}\right )}}\,{d x} + \frac{q^{2} + 3 \, q \log \left (-a x^{q} + 1\right ) - 9 \,{\rm Li}_2\left (a x^{q}\right )}{27 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-q^2*integrate(1/9/(a*x^4*x^q - x^4), x) + 1/27*(q^2 + 3*q*log(-a*x^q + 1) - 9*dilog(a*x^q))/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (a x^{q}\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(dilog(a*x^q)/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (a x^{q}\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog(a*x^q)/x^4, x)