Optimal. Leaf size=86 \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (a x+1)^3}+\frac{3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (a x+1)}+\frac{9 \sqrt{1-a^2 x^2}}{2 a^2}+\frac{9 \sin ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.364178, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6124, 1633, 1593, 12, 793, 665, 216} \[ \frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (a x+1)^3}+\frac{3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (a x+1)}+\frac{9 \sqrt{1-a^2 x^2}}{2 a^2}+\frac{9 \sin ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 6124
Rule 1633
Rule 1593
Rule 12
Rule 793
Rule 665
Rule 216
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a x)} x \, dx &=\int \frac{x (1-a x)^2}{(1+a x) \sqrt{1-a^2 x^2}} \, dx\\ &=a \int \frac{\left (\frac{x}{a}-x^2\right ) \sqrt{1-a^2 x^2}}{(1+a x)^2} \, dx\\ &=a \int \frac{\left (\frac{1}{a}-x\right ) x \sqrt{1-a^2 x^2}}{(1+a x)^2} \, dx\\ &=a^2 \int \frac{x \left (1-a^2 x^2\right )^{3/2}}{a^2 (1+a x)^3} \, dx\\ &=\int \frac{x \left (1-a^2 x^2\right )^{3/2}}{(1+a x)^3} \, dx\\ &=\frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (1+a x)^3}+\frac{3 \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(1+a x)^2} \, dx}{a}\\ &=\frac{3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1+a x)}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (1+a x)^3}+\frac{9 \int \frac{\sqrt{1-a^2 x^2}}{1+a x} \, dx}{2 a}\\ &=\frac{9 \sqrt{1-a^2 x^2}}{2 a^2}+\frac{3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1+a x)}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (1+a x)^3}+\frac{9 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{9 \sqrt{1-a^2 x^2}}{2 a^2}+\frac{3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1+a x)}+\frac{\left (1-a^2 x^2\right )^{5/2}}{a^2 (1+a x)^3}+\frac{9 \sin ^{-1}(a x)}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0533774, size = 44, normalized size = 0.51 \[ \frac{\sqrt{1-a^2 x^2} \left (-a x+\frac{8}{a x+1}+6\right )+9 \sin ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 169, normalized size = 2. \begin{align*} 3\,{\frac{ \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{5/2}}{{a}^{4} \left ( x+{a}^{-1} \right ) ^{2}}}+3\,{\frac{ \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{3/2}}{{a}^{2}}}+{\frac{9\,x}{2\,a}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{9}{2\,a}\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}}}}}+{\frac{1}{{a}^{5} \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46216, size = 149, normalized size = 1.73 \begin{align*} -\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a^{4} x^{2} + 2 \, a^{3} x + a^{2}} + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{2 \,{\left (a^{3} x + a^{2}\right )}} + \frac{6 \, \sqrt{-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac{9 \, \arcsin \left (a x\right )}{2 \, a^{2}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1}}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92971, size = 177, normalized size = 2.06 \begin{align*} \frac{14 \, a x - 18 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (a^{2} x^{2} - 5 \, a x - 14\right )} \sqrt{-a^{2} x^{2} + 1} + 14}{2 \,{\left (a^{3} x + a^{2}\right )}} \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 \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\left (a x + 1\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17709, size = 105, normalized size = 1.22 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{x}{a} - \frac{6}{a^{2}}\right )} + \frac{9 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \, a{\left | a \right |}} - \frac{8}{a{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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