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