Optimal. Leaf size=69 \[ -\frac{a q^2 x^{q-1} \text{Hypergeometric2F1}\left (1,-\frac{1-q}{q},2-\frac{1}{q},a x^q\right )}{1-q}-\frac{\text{PolyLog}\left (2,a x^q\right )}{x}+\frac{q \log \left (1-a x^q\right )}{x} \]
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Rubi [A] time = 0.0393505, antiderivative size = 69, 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} \[ -\frac{\text{PolyLog}\left (2,a x^q\right )}{x}-\frac{a q^2 x^{q-1} \, _2F_1\left (1,-\frac{1-q}{q};2-\frac{1}{q};a x^q\right )}{1-q}+\frac{q \log \left (1-a x^q\right )}{x} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 364
Rubi steps
\begin{align*} \int \frac{\text{Li}_2\left (a x^q\right )}{x^2} \, dx &=-\frac{\text{Li}_2\left (a x^q\right )}{x}-q \int \frac{\log \left (1-a x^q\right )}{x^2} \, dx\\ &=\frac{q \log \left (1-a x^q\right )}{x}-\frac{\text{Li}_2\left (a x^q\right )}{x}+\left (a q^2\right ) \int \frac{x^{-2+q}}{1-a x^q} \, dx\\ &=-\frac{a q^2 x^{-1+q} \, _2F_1\left (1,-\frac{1-q}{q};2-\frac{1}{q};a x^q\right )}{1-q}+\frac{q \log \left (1-a x^q\right )}{x}-\frac{\text{Li}_2\left (a x^q\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.052905, size = 60, normalized size = 0.87 \[ \frac{q \left (\frac{a q x^q \text{Hypergeometric2F1}\left (1,\frac{q-1}{q},2-\frac{1}{q},a x^q\right )}{q-1}+\log \left (1-a x^q\right )\right )}{x}-\frac{\text{PolyLog}\left (2,a x^q\right )}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.208, size = 106, normalized size = 1.5 \begin{align*} -{\frac{\sqrt [q]{-a}}{q} \left ( -{\frac{{q}^{2}\ln \left ( 1-a{x}^{q} \right ) }{x} \left ( -a \right ) ^{-{q}^{-1}}}-{\frac{ \left ( 1-q \right ) q{\it polylog} \left ( 2,a{x}^{q} \right ) }{ \left ( -1+q \right ) x} \left ( -a \right ) ^{-{q}^{-1}}}-{q}^{2}{x}^{-1+q}a \left ( -a \right ) ^{-{q}^{-1}}{\it LerchPhi} \left ( a{x}^{q},1,{\frac{-1+q}{q}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -q^{2} \int \frac{1}{a x^{2} x^{q} - x^{2}}\,{d x} + \frac{q^{2} + q \log \left (-a x^{q} + 1\right ) -{\rm Li}_2\left (a x^{q}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{2}\left (a x^{q}\right )}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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