Optimal. Leaf size=54 \[ \frac{a q^2 x^{q+1} \text{Hypergeometric2F1}\left (1,\frac{1}{q}+1,\frac{1}{q}+2,a x^q\right )}{q+1}+x \text{PolyLog}\left (2,a x^q\right )+q x \log \left (1-a x^q\right ) \]
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Rubi [A] time = 0.023219, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6586, 2448, 364} \[ x \text{PolyLog}\left (2,a x^q\right )+\frac{a q^2 x^{q+1} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{q+1}+q x \log \left (1-a x^q\right ) \]
Antiderivative was successfully verified.
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Rule 6586
Rule 2448
Rule 364
Rubi steps
\begin{align*} \int \text{Li}_2\left (a x^q\right ) \, dx &=x \text{Li}_2\left (a x^q\right )+q \int \log \left (1-a x^q\right ) \, dx\\ &=q x \log \left (1-a x^q\right )+x \text{Li}_2\left (a x^q\right )+\left (a q^2\right ) \int \frac{x^q}{1-a x^q} \, dx\\ &=\frac{a q^2 x^{1+q} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{1+q}+q x \log \left (1-a x^q\right )+x \text{Li}_2\left (a x^q\right )\\ \end{align*}
Mathematica [A] time = 0.0450111, size = 51, normalized size = 0.94 \[ q x \left (\frac{a q x^q \text{Hypergeometric2F1}\left (1,\frac{1}{q}+1,\frac{1}{q}+2,a x^q\right )}{q+1}+\log \left (1-a x^q\right )\right )+x \text{PolyLog}\left (2,a x^q\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.207, size = 88, normalized size = 1.6 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-{q}^{-1}} \left ( -{q}^{2}x\sqrt [q]{-a}\ln \left ( 1-a{x}^{q} \right ) -qx\sqrt [q]{-a}{\it polylog} \left ( 2,a{x}^{q} \right ) -{q}^{2}{x}^{1+q}a\sqrt [q]{-a}{\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} x - q^{2} \int \frac{1}{a x^{q} - 1}\,{d x} + q x \log \left (-a x^{q} + 1\right ) + x{\rm Li}_2\left (a x^{q}\right ) \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 ({\rm Li}_2\left (a x^{q}\right ), 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 \operatorname{Li}_{2}\left (a x^{q}\right )\, 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{\rm Li}_2\left (a x^{q}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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