Optimal. Leaf size=69 \[ -\frac {a q^3 x^{1+q} \, _2F_1\left (1,1+\frac {1}{q};2+\frac {1}{q};a x^q\right )}{1+q}-q^2 x \log \left (1-a x^q\right )-q x \text {PolyLog}\left (2,a x^q\right )+x \text {PolyLog}\left (3,a x^q\right ) \]
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Rubi [A]
time = 0.02, 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 = {6721, 2498, 371}
\begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2498
Rule 6721
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] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 0.01, size = 39, normalized size = 0.57 \begin {gather*} -\frac {x G_{5,5}^{1,5}\left (-a x^q|\begin {array}{c} 1,1,1,1,\frac {-1+q}{q} \\ 1,0,0,0,-\frac {1}{q} \\\end {array}\right )}{q} \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 = 0.19, size = 105, normalized size = 1.52
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{-\frac {1}{q}} \left (q^{3} x \left (-a \right )^{\frac {1}{q}} \ln \left (1-a \,x^{q}\right )+q^{2} x \left (-a \right )^{\frac {1}{q}} \polylog \left (2, a \,x^{q}\right )-q x \left (-a \right )^{\frac {1}{q}} \polylog \left (3, a \,x^{q}\right )+q^{3} x^{1+q} a \left (-a \right )^{\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {1+q}{q}\right )\right )}{q}\) | \(105\) |
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}_{3}\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 = -0.01 \begin {gather*} \int \mathrm {polylog}\left (3,a\,x^q\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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