Optimal. Leaf size=139 \[ \frac{40 x}{27 \sqrt{1-a^2 x^2}}+\frac{2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^2}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{4 \tanh ^{-1}(a x)}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0928308, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 191, 192} \[ \frac{40 x}{27 \sqrt{1-a^2 x^2}}+\frac{2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^2}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{4 \tanh ^{-1}(a x)}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5964
Rule 5962
Rule 191
Rule 192
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2}{9} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{27 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac{4 \tanh ^{-1}(a x)}{3 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^2}{3 \sqrt{1-a^2 x^2}}+\frac{4}{27} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{4}{3} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x}{27 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac{4 \tanh ^{-1}(a x)}{3 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^2}{3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.073697, size = 70, normalized size = 0.5 \[ \frac{-40 a^3 x^3-9 a x \left (2 a^2 x^2-3\right ) \tanh ^{-1}(a x)^2+6 \left (6 a^2 x^2-7\right ) \tanh ^{-1}(a x)+42 a x}{27 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.17, size = 84, normalized size = 0.6 \begin{align*} -{\frac{18\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+40\,{x}^{3}{a}^{3}-36\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -27\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}xa-42\,ax+42\,{\it Artanh} \left ( ax \right ) }{27\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.6954, size = 410, normalized size = 2.95 \begin{align*} \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\right )} \operatorname{artanh}\left (a x\right )^{2} + \frac{1}{27} \, a{\left (\frac{\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x + \sqrt{-a^{2} x^{2} + 1} a}}{a} + \frac{\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x - \sqrt{-a^{2} x^{2} + 1} a}}{a} - \frac{18 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac{18 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac{18 \, \log \left (a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{18 \, \log \left (-a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{3 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} + \frac{3 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.67599, size = 238, normalized size = 1.71 \begin{align*} -\frac{{\left (160 \, a^{3} x^{3} + 9 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 168 \, a x - 12 \,{\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{108 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + 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 \frac{\operatorname{atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, 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 \frac{\operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]