Optimal. Leaf size=95 \[ -\frac {11 \sin ^{-1}(a x)}{2 a^3}-\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\frac {(3-a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {(28-3 a x) \sqrt {1-a^2 x^2}}{6 a^3} \]
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Rubi [A] time = 0.64, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6124, 1633, 1593, 12, 852, 1635, 1654, 780, 216} \[ -\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\frac {(3-a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {(28-3 a x) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {11 \sin ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 780
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1654
Rule 6124
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-a x)^2}{(1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=a \int \frac {\sqrt {1-a^2 x^2} \left (\frac {x^2}{a}-x^3\right )}{(1+a x)^2} \, dx\\ &=a \int \frac {\left (\frac {1}{a}-x\right ) x^2 \sqrt {1-a^2 x^2}}{(1+a x)^2} \, dx\\ &=a^2 \int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{a^2 (1+a x)^3} \, dx\\ &=\int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{(1+a x)^3} \, dx\\ &=\int \frac {x^2 (1-a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\int \frac {\left (\frac {3}{a^2}-\frac {x}{a}\right ) (1-a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\frac {(3-a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}+\frac {1}{3} \int \frac {\left (\frac {3}{a^2}-\frac {x}{a}\right ) (-5+3 a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\frac {(28-3 a x) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {(3-a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {11 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac {(1-a x)^3}{a^3 \sqrt {1-a^2 x^2}}-\frac {(28-3 a x) \sqrt {1-a^2 x^2}}{6 a^3}-\frac {(3-a x)^2 \sqrt {1-a^2 x^2}}{3 a^3}-\frac {11 \sin ^{-1}(a x)}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 58, normalized size = 0.61 \[ -\frac {\frac {\sqrt {1-a^2 x^2} \left (2 a^3 x^3-7 a^2 x^2+19 a x+52\right )}{a x+1}+33 \sin ^{-1}(a x)}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 83, normalized size = 0.87 \[ -\frac {52 \, a x - 66 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 \, a x + 52\right )} \sqrt {-a^{2} x^{2} + 1} + 52}{6 \, {\left (a^{4} x + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 87, normalized size = 0.92 \[ -\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left (x {\left (\frac {2 \, x}{a} - \frac {9}{a^{2}}\right )} + \frac {28}{a^{3}}\right )} - \frac {11 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a^{2} {\left | a \right |}} + \frac {8}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 170, normalized size = 1.79 \[ -\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{6} \left (x +\frac {1}{a}\right )^{3}}-\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{5} \left (x +\frac {1}{a}\right )^{2}}-\frac {11 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{3}}-\frac {11 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2 a^{2}}-\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.43, size = 177, normalized size = 1.86 \[ \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} x^{2} + 2 \, a^{4} x + a^{3}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4} x + a^{3}} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4} x + a^{3}} + \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} x}{2 \, a^{2}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} - \frac {i \, \arcsin \left (a x + 2\right )}{2 \, a^{3}} - \frac {6 \, \arcsin \left (a x\right )}{a^{3}} + \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3}}{a^{3}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 141, normalized size = 1.48 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,a\,\sqrt {-a^2}}-\frac {4\,\sqrt {-a^2}}{a^3}+\frac {a\,x^2}{3\,\sqrt {-a^2}}+\frac {3\,x\,\sqrt {-a^2}}{2\,a^2}\right )}{\sqrt {-a^2}}-\frac {11\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}}+\frac {4\,\sqrt {1-a^2\,x^2}}{a^2\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \int \frac {x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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