Optimal. Leaf size=88 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{a^2 (1-a x)^3}+\frac {3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1-a x)}+\frac {9 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {9 \sin ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.38, antiderivative size = 88, 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} \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{a^2 (1-a x)^3}+\frac {3 \left (1-a^2 x^2\right )^{3/2}}{2 a^2 (1-a x)}+\frac {9 \sqrt {1-a^2 x^2}}{2 a^2}-\frac {9 \sin ^{-1}(a x)}{2 a^2} \]
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\\ &=-\left (a \int \frac {\left (-\frac {x}{a}-x^2\right ) \sqrt {1-a^2 x^2}}{(1-a x)^2} \, dx\right )\\ &=-\left (a \int \frac {\left (-\frac {1}{a}-x\right ) x \sqrt {1-a^2 x^2}}{(1-a x)^2} \, dx\right )\\ &=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 = 0.04, size = 53, normalized size = 0.60 \[ \sqrt {1-a^2 x^2} \left (-\frac {4}{a^2 (a x-1)}+\frac {3}{a^2}+\frac {x}{2 a}\right )-\frac {9 \sin ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 76, normalized size = 0.86 \[ \frac {14 \, a x + 18 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a^{2} x^{2} + 5 \, a x - 14\right )} \sqrt {-a^{2} x^{2} + 1} - 14}{2 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 78, normalized size = 0.89 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a} + \frac {6}{a^{2}}\right )} - \frac {9 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a {\left | a \right |}} + \frac {8}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 102, normalized size = 1.16 \[ -\frac {a \,x^{3}}{2 \sqrt {-a^{2} x^{2}+1}}+\frac {9 x}{2 a \sqrt {-a^{2} x^{2}+1}}-\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \sqrt {a^{2}}}-\frac {3 x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {7}{a^{2} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 80, normalized size = 0.91 \[ -\frac {a x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {9 \, x}{2 \, \sqrt {-a^{2} x^{2} + 1} a} - \frac {9 \, \arcsin \left (a x\right )}{2 \, a^{2}} + \frac {7}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 102, normalized size = 1.16 \[ -\frac {\left (\frac {3}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a}\right )\,\sqrt {1-a^2\,x^2}+\frac {9\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a}-\frac {4\,\sqrt {1-a^2\,x^2}}{a\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}}{\sqrt {-a^2}} \]
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 \[ \int \frac {x \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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