Optimal. Leaf size=54 \[ \frac {a q^2 x^{1+q} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{1+q}+q x \log \left (1-a x^q\right )+x \text {PolyLog}\left (2,a x^q\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6721, 2498, 371}
\begin {gather*} \frac {a q^2 x^{q+1} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{q+1}+x \text {Li}_2\left (a x^q\right )+q x \log \left (1-a x^q\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2498
Rule 6721
Rubi steps
\begin {align*} \int \text {Li}_2\left (a x^q\right ) \, dx &=x \text {Li}_2\left (a x^q\right )+q \int \log \left (1-a x^q\right ) \, dx\\ &=q x \log \left (1-a x^q\right )+x \text {Li}_2\left (a x^q\right )+\left (a q^2\right ) \int \frac {x^q}{1-a x^q} \, dx\\ &=\frac {a q^2 x^{1+q} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{1+q}+q x \log \left (1-a x^q\right )+x \text {Li}_2\left (a x^q\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 51, normalized size = 0.94 \begin {gather*} q x \left (\frac {a q x^q \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{1+q}+\log \left (1-a x^q\right )\right )+x \text {PolyLog}\left (2,a x^q\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.11, size = 88, normalized size = 1.63
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{-\frac {1}{q}} \left (-q^{2} x \left (-a \right )^{\frac {1}{q}} \ln \left (1-a \,x^{q}\right )-q x \left (-a \right )^{\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )-q^{2} x^{1+q} a \left (-a \right )^{\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {1+q}{q}\right )\right )}{q}\) | \(88\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \operatorname {Li}_{2}\left (a x^{q}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {polylog}\left (2,a\,x^q\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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