Optimal. Leaf size=78 \[ -\frac {a q^2 x^{q-2} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}+\frac {q \log \left (1-a x^q\right )}{4 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, 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} \[ -\frac {\text {PolyLog}\left (2,a x^q\right )}{2 x^2}-\frac {a q^2 x^{q-2} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}+\frac {q \log \left (1-a x^q\right )}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2455
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^q\right )}{x^3} \, dx &=-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}-\frac {1}{2} q \int \frac {\log \left (1-a x^q\right )}{x^3} \, dx\\ &=\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}+\frac {1}{4} \left (a q^2\right ) \int \frac {x^{-3+q}}{1-a x^q} \, dx\\ &=-\frac {a q^2 x^{-2+q} \, _2F_1\left (1,-\frac {2-q}{q};2 \left (1-\frac {1}{q}\right );a x^q\right )}{4 (2-q)}+\frac {q \log \left (1-a x^q\right )}{4 x^2}-\frac {\text {Li}_2\left (a x^q\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.78 \[ \frac {q \left (\frac {a q x^q \, _2F_1\left (1,\frac {q-2}{q};2-\frac {2}{q};a x^q\right )}{q-2}+\log \left (1-a x^q\right )\right )-2 \text {Li}_2\left (a x^q\right )}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (a x^{q}\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 108, normalized size = 1.38 \[ -\frac {\left (-a \right )^{\frac {2}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {2}{q}} \ln \left (1-a \,x^{q}\right )}{4 x^{2}}-\frac {q \left (-a \right )^{-\frac {2}{q}} \left (1-\frac {q}{2}\right ) \polylog \left (2, a \,x^{q}\right )}{\left (-2+q \right ) x^{2}}-\frac {q^{2} x^{-2+q} a \left (-a \right )^{-\frac {2}{q}} \Phi \left (a \,x^{q}, 1, \frac {-2+q}{q}\right )}{4}\right )}{q} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -q^{2} \int \frac {1}{4 \, {\left (a x^{3} x^{q} - x^{3}\right )}}\,{d x} + \frac {q^{2} + 2 \, q \log \left (-a x^{q} + 1\right ) - 4 \, {\rm Li}_2\left (a x^{q}\right )}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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