Optimal. Leaf size=29 \[ -\frac {\log (1-a x)}{a^3}-\frac {x}{a^2}-\frac {x^2}{2 a} \]
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Rubi [A] time = 0.09, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 43} \[ -\frac {x}{a^2}-\frac {\log (1-a x)}{a^3}-\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x^2}{1-a x} \, dx\\ &=\int \left (-\frac {1}{a^2}-\frac {x}{a}-\frac {1}{a^2 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{a^2}-\frac {x^2}{2 a}-\frac {\log (1-a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \[ -\frac {\log (1-a x)}{a^3}-\frac {x}{a^2}-\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 25, normalized size = 0.86 \[ -\frac {a^{2} x^{2} + 2 \, a x + 2 \, \log \left (a x - 1\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 27, normalized size = 0.93 \[ -\frac {a x^{2} + 2 \, x}{2 \, a^{2}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 0.93 \[ -\frac {x^{2}}{2 a}-\frac {x}{a^{2}}-\frac {\ln \left (a x -1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 26, normalized size = 0.90 \[ -\frac {a x^{2} + 2 \, x}{2 \, a^{2}} - \frac {\log \left (a x - 1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.87, size = 23, normalized size = 0.79 \[ -\frac {\ln \left (a\,x-1\right )+a\,x+\frac {a^2\,x^2}{2}}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 22, normalized size = 0.76 \[ - \frac {x^{2}}{2 a} - \frac {x}{a^{2}} - \frac {\log {\left (a x - 1 \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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