Optimal. Leaf size=69 \[ -\frac{a q^3 x^{q+1} \text{Hypergeometric2F1}\left (1,\frac{1}{q}+1,\frac{1}{q}+2,a x^q\right )}{q+1}-q x \text{PolyLog}\left (2,a x^q\right )+x \text{PolyLog}\left (3,a x^q\right )-q^2 x \log \left (1-a x^q\right ) \]
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Rubi [A] time = 0.0280803, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6586, 2448, 364} \[ -q x \text{PolyLog}\left (2,a x^q\right )+x \text{PolyLog}\left (3,a x^q\right )-\frac{a q^3 x^{q+1} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{q+1}-q^2 x \log \left (1-a x^q\right ) \]
Antiderivative was successfully verified.
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Rule 6586
Rule 2448
Rule 364
Rubi steps
\begin{align*} \int \text{Li}_3\left (a x^q\right ) \, dx &=x \text{Li}_3\left (a x^q\right )-q \int \text{Li}_2\left (a x^q\right ) \, dx\\ &=-q x \text{Li}_2\left (a x^q\right )+x \text{Li}_3\left (a x^q\right )-q^2 \int \log \left (1-a x^q\right ) \, dx\\ &=-q^2 x \log \left (1-a x^q\right )-q x \text{Li}_2\left (a x^q\right )+x \text{Li}_3\left (a x^q\right )-\left (a q^3\right ) \int \frac{x^q}{1-a x^q} \, dx\\ &=-\frac{a q^3 x^{1+q} \, _2F_1\left (1,1+\frac{1}{q};2+\frac{1}{q};a x^q\right )}{1+q}-q^2 x \log \left (1-a x^q\right )-q x \text{Li}_2\left (a x^q\right )+x \text{Li}_3\left (a x^q\right )\\ \end{align*}
Mathematica [C] time = 0.0060684, size = 39, normalized size = 0.57 \[ -\frac{x G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,\frac{q-1}{q} \\ 1,0,0,0,-\frac{1}{q} \\\end{array}\right )}{q} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.334, size = 105, normalized size = 1.5 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{-{q}^{-1}} \left ({q}^{3}x\sqrt [q]{-a}\ln \left ( 1-a{x}^{q} \right ) +{q}^{2}x\sqrt [q]{-a}{\it polylog} \left ( 2,a{x}^{q} \right ) -qx\sqrt [q]{-a}{\it polylog} \left ( 3,a{x}^{q} \right ) +{q}^{3}{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^{3} x + q^{3} \int \frac{1}{a x^{q} - 1}\,{d x} - q^{2} x \log \left (-a x^{q} + 1\right ) - q x{\rm Li}_2\left (a x^{q}\right ) + x{\rm Li}_{3}(a x^{q}) \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 polylog}\left (3, 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}_{3}\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}_{3}(a x^{q})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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