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 = 0.36, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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 12
Rule 216
Rule 665
Rule 793
Rule 1593
Rule 1633
Rule 6124
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.56, size = 75, normalized size = 0.87 \[ \frac {14 \, a x - 18 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (a^{2} x^{2} - 5 \, a x - 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{2 \, {\left (a^{3} x + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 78, normalized size = 0.91 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a} - \frac {6}{a^{2}}\right )} + \frac {9 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a {\left | a \right |}} - \frac {8}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 169, normalized size = 1.97 \[ \frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{5} \left (x +\frac {1}{a}\right )^{3}}+\frac {3 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{2}}+\frac {3 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a^{2}}+\frac {9 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2 a}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 a \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 110, normalized size = 1.28 \[ -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4} x^{2} + 2 \, a^{3} x + a^{2}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (a^{3} x + a^{2}\right )}} + \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac {9 \, \arcsin \left (a x\right )}{2 \, a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 101, normalized size = 1.17 \[ -\frac {\left (\frac {3}{\sqrt {-a^2}}+\frac {x\,\sqrt {-a^2}}{2\,a}\right )\,\sqrt {1-a^2\,x^2}-\frac {9\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a}+\frac {4\,\sqrt {1-a^2\,x^2}}{a\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )}}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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