Optimal. Leaf size=50 \[ -2 x \text{PolyLog}\left (2,a x^2\right )+x \text{PolyLog}\left (3,a x^2\right )-4 x \log \left (1-a x^2\right )-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{\sqrt{a}}+8 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0233193, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6586, 2448, 321, 206} \[ -2 x \text{PolyLog}\left (2,a x^2\right )+x \text{PolyLog}\left (3,a x^2\right )-4 x \log \left (1-a x^2\right )-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{\sqrt{a}}+8 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6586
Rule 2448
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \text{Li}_3\left (a x^2\right ) \, dx &=x \text{Li}_3\left (a x^2\right )-2 \int \text{Li}_2\left (a x^2\right ) \, dx\\ &=-2 x \text{Li}_2\left (a x^2\right )+x \text{Li}_3\left (a x^2\right )-4 \int \log \left (1-a x^2\right ) \, dx\\ &=-4 x \log \left (1-a x^2\right )-2 x \text{Li}_2\left (a x^2\right )+x \text{Li}_3\left (a x^2\right )-(8 a) \int \frac{x^2}{1-a x^2} \, dx\\ &=8 x-4 x \log \left (1-a x^2\right )-2 x \text{Li}_2\left (a x^2\right )+x \text{Li}_3\left (a x^2\right )-8 \int \frac{1}{1-a x^2} \, dx\\ &=8 x-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{\sqrt{a}}-4 x \log \left (1-a x^2\right )-2 x \text{Li}_2\left (a x^2\right )+x \text{Li}_3\left (a x^2\right )\\ \end{align*}
Mathematica [A] time = 0.087684, size = 50, normalized size = 1. \[ -2 x \text{PolyLog}\left (2,a x^2\right )+x \text{PolyLog}\left (3,a x^2\right )-4 x \log \left (1-a x^2\right )-\frac{8 \tanh ^{-1}\left (\sqrt{a} x\right )}{\sqrt{a}}+8 x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.173, size = 119, normalized size = 2.4 \begin{align*} -{\frac{1}{2} \left ( 16\,{\frac{x \left ( -a \right ) ^{3/2}}{a}}+8\,{\frac{x \left ( -a \right ) ^{3/2} \left ( \ln \left ( 1-\sqrt{a{x}^{2}} \right ) -\ln \left ( 1+\sqrt{a{x}^{2}} \right ) \right ) }{a\sqrt{a{x}^{2}}}}-8\,{\frac{x \left ( -a \right ) ^{3/2}\ln \left ( -a{x}^{2}+1 \right ) }{a}}-4\,{\frac{x \left ( -a \right ) ^{3/2}{\it polylog} \left ( 2,a{x}^{2} \right ) }{a}}+2\,{\frac{x \left ( -a \right ) ^{3/2}{\it polylog} \left ( 3,a{x}^{2} \right ) }{a}} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.69418, size = 452, normalized size = 9.04 \begin{align*} \left [-\frac{2 \, a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) + 4 \, a x \log \left (-a x^{2} + 1\right ) - a x{\rm polylog}\left (3, a x^{2}\right ) - 8 \, a x - 4 \, \sqrt{a} \log \left (\frac{a x^{2} - 2 \, \sqrt{a} x + 1}{a x^{2} - 1}\right )}{a}, -\frac{2 \, a x{\rm \%iint}\left (a, x, -\frac{\log \left (-a x^{2} + 1\right )}{a}, -\frac{2 \, \log \left (-a x^{2} + 1\right )}{x}\right ) + 4 \, a x \log \left (-a x^{2} + 1\right ) - a x{\rm polylog}\left (3, a x^{2}\right ) - 8 \, a x - 8 \, \sqrt{-a} \arctan \left (\sqrt{-a} x\right )}{a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{3}\left (a x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{3}(a x^{2})\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]