Optimal. Leaf size=37 \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{6 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.020476, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6097, 260} \[ \frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{6 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6097
Rule 260
Rubi steps
\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )-(b c) \int \frac{x^5}{1-c^2 x^6} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )+\frac{b \log \left (1-c^2 x^6\right )}{6 c}\\ \end{align*}
Mathematica [A] time = 0.008322, size = 42, normalized size = 1.14 \[ \frac{a x^3}{3}+\frac{b \log \left (1-c^2 x^6\right )}{6 c}+\frac{1}{3} b x^3 \tanh ^{-1}\left (c x^3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 37, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}{\it Artanh} \left ( c{x}^{3} \right ) }{3}}+{\frac{b\ln \left ( -{c}^{2}{x}^{6}+1 \right ) }{6\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01133, size = 50, normalized size = 1.35 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{{\left (2 \, c x^{3} \operatorname{artanh}\left (c x^{3}\right ) + \log \left (-c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9613, size = 108, normalized size = 2.92 \begin{align*} \frac{b c x^{3} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 2 \, a c x^{3} + b \log \left (c^{2} x^{6} - 1\right )}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1694, size = 66, normalized size = 1.78 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{6} \,{\left (x^{3} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + \frac{\log \left ({\left | c^{2} x^{6} - 1 \right |}\right )}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]