Optimal. Leaf size=142 \[ \frac{81 a (d x)^{m+4} \text{Hypergeometric2F1}\left (1,\frac{m+4}{3},\frac{m+7}{3},a x^3\right )}{d^4 (m+1)^4 (m+4)}+\frac{9 (d x)^{m+1} \text{PolyLog}\left (2,a x^3\right )}{d (m+1)^3}-\frac{3 (d x)^{m+1} \text{PolyLog}\left (3,a x^3\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (4,a x^3\right )}{d (m+1)}+\frac{27 \log \left (1-a x^3\right ) (d x)^{m+1}}{d (m+1)^4} \]
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Rubi [A] time = 0.0943941, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2455, 16, 364} \[ \frac{9 (d x)^{m+1} \text{PolyLog}\left (2,a x^3\right )}{d (m+1)^3}-\frac{3 (d x)^{m+1} \text{PolyLog}\left (3,a x^3\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (4,a x^3\right )}{d (m+1)}+\frac{81 a (d x)^{m+4} \, _2F_1\left (1,\frac{m+4}{3};\frac{m+7}{3};a x^3\right )}{d^4 (m+1)^4 (m+4)}+\frac{27 \log \left (1-a x^3\right ) (d x)^{m+1}}{d (m+1)^4} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2455
Rule 16
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \text{Li}_4\left (a x^3\right ) \, dx &=\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}-\frac{3 \int (d x)^m \text{Li}_3\left (a x^3\right ) \, dx}{1+m}\\ &=-\frac{3 (d x)^{1+m} \text{Li}_3\left (a x^3\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}+\frac{9 \int (d x)^m \text{Li}_2\left (a x^3\right ) \, dx}{(1+m)^2}\\ &=\frac{9 (d x)^{1+m} \text{Li}_2\left (a x^3\right )}{d (1+m)^3}-\frac{3 (d x)^{1+m} \text{Li}_3\left (a x^3\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}+\frac{27 \int (d x)^m \log \left (1-a x^3\right ) \, dx}{(1+m)^3}\\ &=\frac{27 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^4}+\frac{9 (d x)^{1+m} \text{Li}_2\left (a x^3\right )}{d (1+m)^3}-\frac{3 (d x)^{1+m} \text{Li}_3\left (a x^3\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}+\frac{(81 a) \int \frac{x^2 (d x)^{1+m}}{1-a x^3} \, dx}{d (1+m)^4}\\ &=\frac{27 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^4}+\frac{9 (d x)^{1+m} \text{Li}_2\left (a x^3\right )}{d (1+m)^3}-\frac{3 (d x)^{1+m} \text{Li}_3\left (a x^3\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}+\frac{(81 a) \int \frac{(d x)^{3+m}}{1-a x^3} \, dx}{d^3 (1+m)^4}\\ &=\frac{81 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)^4 (4+m)}+\frac{27 (d x)^{1+m} \log \left (1-a x^3\right )}{d (1+m)^4}+\frac{9 (d x)^{1+m} \text{Li}_2\left (a x^3\right )}{d (1+m)^3}-\frac{3 (d x)^{1+m} \text{Li}_3\left (a x^3\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_4\left (a x^3\right )}{d (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0967328, size = 166, normalized size = 1.17 \[ \frac{3 x \text{Gamma}\left (\frac{m+4}{3}\right ) (d x)^m \left (9 a (m+1) x^3 \text{Gamma}\left (\frac{m+1}{3}\right ) \, _2\tilde{F}_1\left (1,\frac{m+4}{3};\frac{m+7}{3};a x^3\right )+m^3 \text{PolyLog}\left (4,a x^3\right )-3 m^2 \text{PolyLog}\left (3,a x^3\right )+3 m^2 \text{PolyLog}\left (4,a x^3\right )-6 m \text{PolyLog}\left (3,a x^3\right )+3 m \text{PolyLog}\left (4,a x^3\right )+9 (m+1) \text{PolyLog}\left (2,a x^3\right )-3 \text{PolyLog}\left (3,a x^3\right )+\text{PolyLog}\left (4,a x^3\right )+27 \log \left (1-a x^3\right )\right )}{(m+1)^5 \text{Gamma}\left (\frac{m+1}{3}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.856, size = 259, normalized size = 1.8 \begin{align*} -{\frac{ \left ( dx \right ) ^{m}{x}^{-m}}{3} \left ( -a \right ) ^{-{\frac{1}{3}}-{\frac{m}{3}}} \left ( 3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3} \left ( -324-81\,m \right ) }{ \left ( 1+m \right ) ^{5} \left ( 4+m \right ) a}}-3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3} \left ( -108-27\,m \right ) \ln \left ( -{x}^{3}a+1 \right ) }{ \left ( 1+m \right ) ^{4} \left ( 4+m \right ) a}}+3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3} \left ( 36+9\,m \right ){\it polylog} \left ( 2,{x}^{3}a \right ) }{ \left ( 1+m \right ) ^{3} \left ( 4+m \right ) a}}+3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3} \left ( -12-3\,m \right ){\it polylog} \left ( 3,{x}^{3}a \right ) }{ \left ( 1+m \right ) ^{2} \left ( 4+m \right ) a}}+3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3}{\it polylog} \left ( 4,{x}^{3}a \right ) }{ \left ( 1+m \right ) a}}+3\,{\frac{{x}^{1+m} \left ( -a \right ) ^{4/3+m/3} \left ( 108+27\,m \right ){\it LerchPhi} \left ({x}^{3}a,1,m/3+1/3 \right ) }{ \left ( 1+m \right ) ^{4} \left ( 4+m \right ) a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -81 \, a d^{m} \int -\frac{x^{3} x^{m}}{m^{4} -{\left (a m^{4} + 4 \, a m^{3} + 6 \, a m^{2} + 4 \, a m + a\right )} x^{3} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1}\,{d x} + \frac{9 \,{\left (d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_2\left (a x^{3}\right ) + 27 \, d^{m} x x^{m} \log \left (-a x^{3} + 1\right ) +{\left (d^{m} m^{3} + 3 \, d^{m} m^{2} + 3 \, d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_{4}(a x^{3}) - 3 \,{\left (d^{m} m^{2} + 2 \, d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_{3}(a x^{3})}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d x\right )^{m}{\rm polylog}\left (4, a x^{3}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \operatorname{Li}_{4}\left (a x^{3}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m}{\rm Li}_{4}(a x^{3})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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