Optimal. Leaf size=118 \[ -\frac{8 a (d x)^{m+3} \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},a x^2\right )}{d^3 (m+1)^3 (m+3)}-\frac{2 (d x)^{m+1} \text{PolyLog}\left (2,a x^2\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (3,a x^2\right )}{d (m+1)}-\frac{4 \log \left (1-a x^2\right ) (d x)^{m+1}}{d (m+1)^3} \]
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Rubi [A] time = 0.0712051, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6591, 2455, 16, 364} \[ -\frac{2 (d x)^{m+1} \text{PolyLog}\left (2,a x^2\right )}{d (m+1)^2}+\frac{(d x)^{m+1} \text{PolyLog}\left (3,a x^2\right )}{d (m+1)}-\frac{8 a (d x)^{m+3} \, _2F_1\left (1,\frac{m+3}{2};\frac{m+5}{2};a x^2\right )}{d^3 (m+1)^3 (m+3)}-\frac{4 \log \left (1-a x^2\right ) (d x)^{m+1}}{d (m+1)^3} \]
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}_3\left (a x^2\right ) \, dx &=\frac{(d x)^{1+m} \text{Li}_3\left (a x^2\right )}{d (1+m)}-\frac{2 \int (d x)^m \text{Li}_2\left (a x^2\right ) \, dx}{1+m}\\ &=-\frac{2 (d x)^{1+m} \text{Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^2\right )}{d (1+m)}-\frac{4 \int (d x)^m \log \left (1-a x^2\right ) \, dx}{(1+m)^2}\\ &=-\frac{4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac{2 (d x)^{1+m} \text{Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^2\right )}{d (1+m)}-\frac{(8 a) \int \frac{x (d x)^{1+m}}{1-a x^2} \, dx}{d (1+m)^3}\\ &=-\frac{4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac{2 (d x)^{1+m} \text{Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^2\right )}{d (1+m)}-\frac{(8 a) \int \frac{(d x)^{2+m}}{1-a x^2} \, dx}{d^2 (1+m)^3}\\ &=-\frac{8 a (d x)^{3+m} \, _2F_1\left (1,\frac{3+m}{2};\frac{5+m}{2};a x^2\right )}{d^3 (1+m)^3 (3+m)}-\frac{4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac{2 (d x)^{1+m} \text{Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_3\left (a x^2\right )}{d (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0792053, size = 126, normalized size = 1.07 \[ -\frac{2 x \text{Gamma}\left (\frac{m+3}{2}\right ) (d x)^m \left (2 a (m+1) x^2 \text{Gamma}\left (\frac{m+1}{2}\right ) \, _2\tilde{F}_1\left (1,\frac{m+3}{2};\frac{m+5}{2};a x^2\right )+m^2 \left (-\text{PolyLog}\left (3,a x^2\right )\right )-2 m \text{PolyLog}\left (3,a x^2\right )+2 (m+1) \text{PolyLog}\left (2,a x^2\right )-\text{PolyLog}\left (3,a x^2\right )+4 \log \left (1-a x^2\right )\right )}{(m+1)^4 \text{Gamma}\left (\frac{m+1}{2}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.41, size = 218, normalized size = 1.9 \begin{align*} -{\frac{ \left ( dx \right ) ^{m}{x}^{-m}}{2} \left ( -a \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}} \left ( 2\,{\frac{{x}^{1+m} \left ( -a \right ) ^{3/2+m/2} \left ( 24+8\,m \right ) }{ \left ( 1+m \right ) ^{4} \left ( 3+m \right ) a}}-2\,{\frac{{x}^{1+m} \left ( -a \right ) ^{3/2+m/2} \left ( 12+4\,m \right ) \ln \left ( -a{x}^{2}+1 \right ) }{ \left ( 1+m \right ) ^{3} \left ( 3+m \right ) a}}+2\,{\frac{{x}^{1+m} \left ( -a \right ) ^{3/2+m/2} \left ( -6-2\,m \right ){\it polylog} \left ( 2,a{x}^{2} \right ) }{ \left ( 1+m \right ) ^{2} \left ( 3+m \right ) a}}+2\,{\frac{{x}^{1+m} \left ( -a \right ) ^{3/2+m/2}{\it polylog} \left ( 3,a{x}^{2} \right ) }{ \left ( 1+m \right ) a}}+2\,{\frac{{x}^{1+m} \left ( -a \right ) ^{3/2+m/2} \left ( -12-4\,m \right ){\it LerchPhi} \left ( a{x}^{2},1,1/2+m/2 \right ) }{ \left ( 1+m \right ) ^{3} \left ( 3+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*} 8 \, a d^{m} \int \frac{x^{2} x^{m}}{{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} a x^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1}\,{d x} - \frac{2 \, d^{m}{\left (m + 1\right )} x x^{m}{\rm Li}_2\left (a x^{2}\right ) -{\left (m^{2} + 2 \, m + 1\right )} d^{m} x x^{m}{\rm Li}_{3}(a x^{2}) + 4 \, d^{m} x x^{m} \log \left (-a x^{2} + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, 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 (3, a x^{2}\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}_{3}\left (a x^{2}\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}_{3}(a x^{2})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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