Optimal. Leaf size=69 \[ -\frac {a q^3 x^{q+1} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{q+1}-q x \text {Li}_2\left (a x^q\right )+x \text {Li}_3\left (a x^q\right )-q^2 x \log \left (1-a x^q\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 364
Rule 2448
Rule 6586
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*}
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Mathematica [C] time = 0.01, 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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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm polylog}\left (3, a x^{q}\right ), x\right ) \]
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 \[ \int {\rm Li}_{3}(a x^{q})\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 105, normalized size = 1.52 \[ -\frac {\left (-a \right )^{-\frac {1}{q}} \left (q^{3} x \left (-a \right )^{\frac {1}{q}} \ln \left (1-a \,x^{q}\right )+q^{2} x \left (-a \right )^{\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )-q x \left (-a \right )^{\frac {1}{q}} \polylog \left (3, a \,x^{q}\right )+q^{3} x^{1+q} a \left (-a \right )^{\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {1+q}{q}\right )\right )}{q} \]
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 \[ q^{3} x + q^{3} \int \frac {1}{a x^{q} - 1}\,{d x} - q^{2} x \log \left (-a x^{q} + 1\right ) - q x {\rm Li}_2\left (a x^{q}\right ) + x {\rm Li}_{3}(a x^{q}) \]
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 \[ \int \mathrm {polylog}\left (3,a\,x^q\right ) \,d x \]
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 \[ \int \operatorname {Li}_{3}\left (a x^{q}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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