Optimal. Leaf size=94 \[ \frac {9 a (d x)^{4+m} \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};a x^3\right )}{d^4 (1+m)^2 (4+m)}+\frac {3 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {PolyLog}\left (2,a x^3\right )}{d (1+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6726, 2505, 16,
371} \begin {gather*} \frac {9 a (d x)^{m+4} \, _2F_1\left (1,\frac {m+4}{3};\frac {m+7}{3};a x^3\right )}{d^4 (m+1)^2 (m+4)}+\frac {\text {Li}_2\left (a x^3\right ) (d x)^{m+1}}{d (m+1)}+\frac {3 \log \left (1-a x^3\right ) (d x)^{m+1}}{d (m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int (d x)^m \text {Li}_2\left (a x^3\right ) \, dx &=\frac {(d x)^{1+m} \text {Li}_2\left (a x^3\right )}{d (1+m)}+\frac {3 \int (d x)^m \log \left (1-a x^3\right ) \, dx}{1+m}\\ &=\frac {3 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2\left (a x^3\right )}{d (1+m)}+\frac {(9 a) \int \frac {x^2 (d x)^{1+m}}{1-a x^3} \, dx}{d (1+m)^2}\\ &=\frac {3 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2\left (a x^3\right )}{d (1+m)}+\frac {(9 a) \int \frac {(d x)^{3+m}}{1-a x^3} \, dx}{d^3 (1+m)^2}\\ &=\frac {9 a (d x)^{4+m} \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};a x^3\right )}{d^4 (1+m)^2 (4+m)}+\frac {3 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2\left (a x^3\right )}{d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 72, normalized size = 0.77 \begin {gather*} \frac {x (d x)^m \left (9 a x^3 \, _2F_1\left (1,\frac {4+m}{3};\frac {7+m}{3};a x^3\right )+(4+m) \left (3 \log \left (1-a x^3\right )+(1+m) \text {PolyLog}\left (2,a x^3\right )\right )\right )}{(1+m)^2 (4+m)} \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 = 0.16, size = 177, normalized size = 1.88
method | result | size |
meijerg | \(-\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-\frac {1}{3}-\frac {m}{3}} \left (\frac {3 x^{1+m} \left (-a \right )^{\frac {4}{3}+\frac {m}{3}} \left (-36-9 m \right )}{\left (4+m \right ) \left (1+m \right )^{3} a}-\frac {3 x^{1+m} \left (-a \right )^{\frac {4}{3}+\frac {m}{3}} \left (-12-3 m \right ) \ln \left (-a \,x^{3}+1\right )}{\left (4+m \right ) \left (1+m \right )^{2} a}+\frac {3 x^{1+m} \left (-a \right )^{\frac {4}{3}+\frac {m}{3}} \polylog \left (2, a \,x^{3}\right )}{\left (1+m \right ) a}+\frac {3 x^{1+m} \left (-a \right )^{\frac {4}{3}+\frac {m}{3}} \left (12+3 m \right ) \Phi \left (a \,x^{3}, 1, \frac {m}{3}+\frac {1}{3}\right )}{\left (4+m \right ) \left (1+m \right )^{2} a}\right )}{3}\) | \(177\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (2,a\,x^3\right )\,{\left (d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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