Optimal. Leaf size=93 \[ -\frac{a q^3 x^{q-3} \text{Hypergeometric2F1}\left (1,-\frac{3-q}{q},2-\frac{3}{q},a x^q\right )}{27 (3-q)}-\frac{q \text{PolyLog}\left (2,a x^q\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^q\right )}{3 x^3}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3} \]
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Rubi [A] time = 0.0508152, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2455, 364} \[ -\frac{q \text{PolyLog}\left (2,a x^q\right )}{9 x^3}-\frac{\text{PolyLog}\left (3,a x^q\right )}{3 x^3}-\frac{a q^3 x^{q-3} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{27 (3-q)}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 364
Rubi steps
\begin{align*} \int \frac{\text{Li}_3\left (a x^q\right )}{x^4} \, dx &=-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}+\frac{1}{3} q \int \frac{\text{Li}_2\left (a x^q\right )}{x^4} \, dx\\ &=-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}-\frac{1}{9} q^2 \int \frac{\log \left (1-a x^q\right )}{x^4} \, dx\\ &=\frac{q^2 \log \left (1-a x^q\right )}{27 x^3}-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}+\frac{1}{27} \left (a q^3\right ) \int \frac{x^{-4+q}}{1-a x^q} \, dx\\ &=-\frac{a q^3 x^{-3+q} \, _2F_1\left (1,-\frac{3-q}{q};2-\frac{3}{q};a x^q\right )}{27 (3-q)}+\frac{q^2 \log \left (1-a x^q\right )}{27 x^3}-\frac{q \text{Li}_2\left (a x^q\right )}{9 x^3}-\frac{\text{Li}_3\left (a x^q\right )}{3 x^3}\\ \end{align*}
Mathematica [C] time = 0.0104082, size = 41, normalized size = 0.44 \[ -\frac{G_{5,5}^{1,5}\left (-a x^q|\begin{array}{c} 1,1,1,1,\frac{q+3}{q} \\ 1,0,0,0,\frac{3}{q} \\\end{array}\right )}{q x^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.34, size = 132, normalized size = 1.4 \begin{align*} -{\frac{1}{q} \left ( -a \right ) ^{3\,{q}^{-1}} \left ( -{\frac{{q}^{3}\ln \left ( 1-a{x}^{q} \right ) }{27\,{x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}}}+{\frac{{q}^{2}{\it polylog} \left ( 2,a{x}^{q} \right ) }{9\,{x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}}}-{\frac{q{\it polylog} \left ( 3,a{x}^{q} \right ) }{ \left ( -3+q \right ){x}^{3}} \left ( -a \right ) ^{-3\,{q}^{-1}} \left ( 1-{\frac{q}{3}} \right ) }-{\frac{{q}^{3}{x}^{-3+q}a}{27} \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^{3} \int \frac{1}{27 \,{\left (a x^{4} x^{q} - x^{4}\right )}}\,{d x} + \frac{q^{3} + 3 \, q^{2} \log \left (-a x^{q} + 1\right ) - 9 \, q{\rm Li}_2\left (a x^{q}\right ) - 27 \,{\rm Li}_{3}(a x^{q})}{81 \, 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 polylog}\left (3, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (a x^{q}\right )}{x^{4}}\, 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}_{3}(a x^{q})}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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