Optimal. Leaf size=84 \[ -\frac {a q^3 x^{-1+q} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q^2 \log \left (1-a x^q\right )}{x}-\frac {q \text {PolyLog}\left (2,a x^q\right )}{x}-\frac {\text {PolyLog}\left (3,a x^q\right )}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {6726, 2505,
371} \begin {gather*} -\frac {a q^3 x^{q-1} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}-\frac {q \text {Li}_2\left (a x^q\right )}{x}-\frac {\text {Li}_3\left (a x^q\right )}{x}+\frac {q^2 \log \left (1-a x^q\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3\left (a x^q\right )}{x^2} \, dx &=-\frac {\text {Li}_3\left (a x^q\right )}{x}+q \int \frac {\text {Li}_2\left (a x^q\right )}{x^2} \, dx\\ &=-\frac {q \text {Li}_2\left (a x^q\right )}{x}-\frac {\text {Li}_3\left (a x^q\right )}{x}-q^2 \int \frac {\log \left (1-a x^q\right )}{x^2} \, dx\\ &=\frac {q^2 \log \left (1-a x^q\right )}{x}-\frac {q \text {Li}_2\left (a x^q\right )}{x}-\frac {\text {Li}_3\left (a x^q\right )}{x}+\left (a q^3\right ) \int \frac {x^{-2+q}}{1-a x^q} \, dx\\ &=-\frac {a q^3 x^{-1+q} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q^2 \log \left (1-a x^q\right )}{x}-\frac {q \text {Li}_2\left (a x^q\right )}{x}-\frac {\text {Li}_3\left (a x^q\right )}{x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 0.01, size = 37, normalized size = 0.44 \begin {gather*} -\frac {G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,1+\frac {1}{q} \\ 1,0,0,0,\frac {1}{q} \\\end {array}\right )}{q x} \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 = 0.21, size = 129, normalized size = 1.54
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{\frac {1}{q}} \left (-\frac {q^{3} \left (-a \right )^{-\frac {1}{q}} \ln \left (1-a \,x^{q}\right )}{x}+\frac {q^{2} \left (-a \right )^{-\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )}{x}-\frac {q \left (-a \right )^{-\frac {1}{q}} \left (1-q \right ) \polylog \left (3, a \,x^{q}\right )}{\left (-1+q \right ) x}-q^{3} x^{-1+q} a \left (-a \right )^{-\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {-1+q}{q}\right )\right )}{q}\) | \(129\) |
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 \frac {\operatorname {Li}_{3}\left (a x^{q}\right )}{x^{2}}\, 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 = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (3,a\,x^q\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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