Optimal. Leaf size=69 \[ -\frac {a q^2 x^{q-1} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}-\frac {\text {Li}_2\left (a x^q\right )}{x}+\frac {q \log \left (1-a x^q\right )}{x} \]
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Rubi [A] time = 0.04, antiderivative size = 69, 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} \[ -\frac {\text {PolyLog}\left (2,a x^q\right )}{x}-\frac {a q^2 x^{q-1} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q \log \left (1-a x^q\right )}{x} \]
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^2} \, dx &=-\frac {\text {Li}_2\left (a x^q\right )}{x}-q \int \frac {\log \left (1-a x^q\right )}{x^2} \, dx\\ &=\frac {q \log \left (1-a x^q\right )}{x}-\frac {\text {Li}_2\left (a x^q\right )}{x}+\left (a q^2\right ) \int \frac {x^{-2+q}}{1-a x^q} \, dx\\ &=-\frac {a q^2 x^{-1+q} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q \log \left (1-a x^q\right )}{x}-\frac {\text {Li}_2\left (a x^q\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.87 \[ \frac {q \left (\frac {a q x^q \, _2F_1\left (1,\frac {q-1}{q};2-\frac {1}{q};a x^q\right )}{q-1}+\log \left (1-a x^q\right )\right )}{x}-\frac {\text {Li}_2\left (a x^q\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm Li}_2\left (a x^{q}\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\rm Li}_2\left (a x^{q}\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 106, normalized size = 1.54 \[ -\frac {\left (-a \right )^{\frac {1}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {1}{q}} \ln \left (1-a \,x^{q}\right )}{x}-\frac {q \left (-a \right )^{-\frac {1}{q}} \left (1-q \right ) \polylog \left (2, a \,x^{q}\right )}{\left (-1+q \right ) x}-q^{2} x^{-1+q} a \left (-a \right )^{-\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {-1+q}{q}\right )\right )}{q} \]
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 \[ -q^{2} \int \frac {1}{a x^{2} x^{q} - x^{2}}\,{d x} + \frac {q^{2} + q \log \left (-a x^{q} + 1\right ) - {\rm Li}_2\left (a x^{q}\right )}{x} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,a\,x^q\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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