Optimal. Leaf size=50 \[ \frac{a^3 x^5}{30}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{6 a^2}-\frac{a x^3}{9}+\frac{x}{6 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0370326, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {5994, 194} \[ \frac{a^3 x^5}{30}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{6 a^2}-\frac{a x^3}{9}+\frac{x}{6 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5994
Rule 194
Rubi steps
\begin{align*} \int x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx &=-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{6 a^2}+\frac{\int \left (1-a^2 x^2\right )^2 \, dx}{6 a}\\ &=-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{6 a^2}+\frac{\int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx}{6 a}\\ &=\frac{x}{6 a}-\frac{a x^3}{9}+\frac{a^3 x^5}{30}-\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{6 a^2}\\ \end{align*}
Mathematica [A] time = 0.0230581, size = 93, normalized size = 1.86 \[ \frac{a^3 x^5}{30}+\frac{1}{6} a^4 x^6 \tanh ^{-1}(a x)-\frac{1}{2} a^2 x^4 \tanh ^{-1}(a x)+\frac{\log (1-a x)}{12 a^2}-\frac{\log (a x+1)}{12 a^2}-\frac{a x^3}{9}+\frac{1}{2} x^2 \tanh ^{-1}(a x)+\frac{x}{6 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 77, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{\it Artanh} \left ( ax \right ){x}^{6}}{6}}-{\frac{{a}^{2}{\it Artanh} \left ( ax \right ){x}^{4}}{2}}+{\frac{{\it Artanh} \left ( ax \right ){x}^{2}}{2}}+{\frac{{x}^{5}{a}^{3}}{30}}-{\frac{{x}^{3}a}{9}}+{\frac{x}{6\,a}}+{\frac{\ln \left ( ax-1 \right ) }{12\,{a}^{2}}}-{\frac{\ln \left ( ax+1 \right ) }{12\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.95139, size = 62, normalized size = 1.24 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )}{6 \, a^{2}} + \frac{3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x}{90 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.90612, size = 154, normalized size = 3.08 \begin{align*} \frac{6 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 30 \, a x + 15 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{180 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.18522, size = 68, normalized size = 1.36 \begin{align*} \begin{cases} \frac{a^{4} x^{6} \operatorname{atanh}{\left (a x \right )}}{6} + \frac{a^{3} x^{5}}{30} - \frac{a^{2} x^{4} \operatorname{atanh}{\left (a x \right )}}{2} - \frac{a x^{3}}{9} + \frac{x^{2} \operatorname{atanh}{\left (a x \right )}}{2} + \frac{x}{6 a} - \frac{\operatorname{atanh}{\left (a x \right )}}{6 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1609, size = 77, normalized size = 1.54 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )}^{3} \log \left (-\frac{a x + 1}{a x - 1}\right )}{12 \, a^{2}} + \frac{3 \, a^{4} x^{5} - 10 \, a^{2} x^{3} + 15 \, x}{90 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]