Optimal. Leaf size=73 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac{x \sqrt{1-a^2 x^2}}{2 a^2}+\frac{\sqrt{1-a^2 x^2}}{a^3}+\frac{\sin ^{-1}(a x)}{2 a^3} \]
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Rubi [A] time = 0.055245, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6124, 797, 641, 195, 216} \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac{x \sqrt{1-a^2 x^2}}{2 a^2}+\frac{\sqrt{1-a^2 x^2}}{a^3}+\frac{\sin ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 6124
Rule 797
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1-a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{\int \frac{1-a x}{\sqrt{1-a^2 x^2}} \, dx}{a^2}-\frac{\int (1-a x) \sqrt{1-a^2 x^2} \, dx}{a^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2}-\frac{\int \sqrt{1-a^2 x^2} \, dx}{a^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{x \sqrt{1-a^2 x^2}}{2 a^2}-\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac{\sin ^{-1}(a x)}{a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{x \sqrt{1-a^2 x^2}}{2 a^2}-\frac{\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac{\sin ^{-1}(a x)}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0316323, size = 43, normalized size = 0.59 \[ \frac{\sqrt{1-a^2 x^2} \left (2 a^2 x^2-3 a x+4\right )+3 \sin ^{-1}(a x)}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 134, normalized size = 1.8 \begin{align*} -{\frac{1}{3\,{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{2\,{a}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{2\,{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{a}^{3}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{1}{{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43345, size = 82, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{3}} + \frac{\arcsin \left (a x\right )}{2 \, a^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85855, size = 131, normalized size = 1.79 \begin{align*} \frac{{\left (2 \, a^{2} x^{2} - 3 \, a x + 4\right )} \sqrt{-a^{2} x^{2} + 1} - 6 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{6 \, a^{3}} \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 \frac{x^{2} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17102, size = 68, normalized size = 0.93 \begin{align*} \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1}{\left (x{\left (\frac{2 \, x}{a} - \frac{3}{a^{2}}\right )} + \frac{4}{a^{3}}\right )} + \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a^{2}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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