Optimal. Leaf size=102 \[ -\frac {a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^3 (m+2)}-\frac {\text {Li}_2(a x) (d x)^{m+1}}{d (m+1)^2}+\frac {\text {Li}_3(a x) (d x)^{m+1}}{d (m+1)}-\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 102, 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 = {6591, 2395, 64} \[ -\frac {(d x)^{m+1} \text {PolyLog}(2,a x)}{d (m+1)^2}+\frac {(d x)^{m+1} \text {PolyLog}(3,a x)}{d (m+1)}-\frac {a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^3 (m+2)}-\frac {\log (1-a x) (d x)^{m+1}}{d (m+1)^3} \]
Antiderivative was successfully verified.
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Rule 64
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int (d x)^m \text {Li}_3(a x) \, dx &=\frac {(d x)^{1+m} \text {Li}_3(a x)}{d (1+m)}-\frac {\int (d x)^m \text {Li}_2(a x) \, dx}{1+m}\\ &=-\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3(a x)}{d (1+m)}-\frac {\int (d x)^m \log (1-a x) \, dx}{(1+m)^2}\\ &=-\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^3}-\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3(a x)}{d (1+m)}-\frac {a \int \frac {(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^3}\\ &=-\frac {a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^3 (2+m)}-\frac {(d x)^{1+m} \log (1-a x)}{d (1+m)^3}-\frac {(d x)^{1+m} \text {Li}_2(a x)}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3(a x)}{d (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 88, normalized size = 0.86 \[ -\frac {x \Gamma (m+2) (d x)^m \left (a (m+1) x \Gamma (m+1) \, _2\tilde {F}_1(1,m+2;m+3;a x)+m^2 (-\text {Li}_3(a x))-2 m \text {Li}_3(a x)+(m+1) \text {Li}_2(a x)-\text {Li}_3(a x)+\log (1-a x)\right )}{(m+1)^4 \Gamma (m+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} {\rm polylog}\left (3, a x\right ), x\right ) \]
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 \[ \int \left (d x\right )^{m} {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 173, normalized size = 1.70 \[ \frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-m} \left (\frac {x^{m} \left (-a \right )^{m} \left (a \,m^{2} x +2 a m x +m^{2}+3 m +2\right )}{\left (m +2\right ) \left (1+m \right )^{4} m}-\frac {x^{1+m} \left (-a \right )^{m} a \ln \left (-a x +1\right )}{\left (1+m \right )^{3}}+\frac {x^{1+m} \left (-a \right )^{m} a \left (-2-m \right ) \polylog \left (2, a x \right )}{\left (m +2\right ) \left (1+m \right )^{2}}+\frac {x^{1+m} \left (-a \right )^{m} a \polylog \left (3, a x \right )}{1+m}+\frac {x^{m} \left (-a \right )^{m} \left (-2-m \right ) \Phi \left (a x , 1, m\right )}{\left (m +2\right ) \left (1+m \right )^{3}}\right )}{a} \]
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 \[ a d^{m} \int -\frac {x x^{m}}{m^{3} - {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a x + 3 \, m^{2} + 3 \, m + 1}\,{d x} - \frac {d^{m} {\left (m + 1\right )} x x^{m} {\rm Li}_2\left (a x\right ) - {\left (m^{2} + 2 \, m + 1\right )} d^{m} x x^{m} {\rm Li}_{3}(a x) + d^{m} x x^{m} \log \left (-a x + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]
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 \[ \int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]
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 \[ \int \left (d x\right )^{m} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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