Optimal. Leaf size=78 \[ \frac {a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^2 (2+m)}+\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {PolyLog}(2,a x)}{d (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6726, 2442, 66}
\begin {gather*} \frac {a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^2 (m+2)}+\frac {\text {Li}_2(a x) (d x)^{m+1}}{d (m+1)}+\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int (d x)^m \text {Li}_2(a x) \, dx &=\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}+\frac {\int (d x)^m \log (1-a x) \, dx}{1+m}\\ &=\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}+\frac {a \int \frac {(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^2}\\ &=\frac {a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^2 (2+m)}+\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.68 \begin {gather*} \frac {x (d x)^m (a x \, _2F_1(1,2+m;3+m;a x)+(2+m) (\log (1-a x)+(1+m) \text {PolyLog}(2,a x)))}{(1+m)^2 (2+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 = 0.12, size = 144, normalized size = 1.85
method | result | size |
meijerg | \(\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (-a \,m^{2} x -2 a m x -m^{2}-3 m -2\right )}{\left (2+m \right ) \left (1+m \right )^{3} m}-\frac {x^{1+m} a \left (-a \right )^{m} \left (-m -2\right ) \ln \left (-a x +1\right )}{\left (2+m \right ) \left (1+m \right )^{2}}+\frac {x^{1+m} a \left (-a \right )^{m} \polylog \left (2, a x \right )}{1+m}+\frac {x^{m} \left (-a \right )^{m} \Phi \left (a x , 1, m\right )}{\left (1+m \right )^{2}}\right )}{a}\) | \(144\) |
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 \left (d x\right )^{m} \operatorname {Li}_{2}\left (a x\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 {\left (d\,x\right )}^m\,\mathrm {polylog}\left (2,a\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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