Optimal. Leaf size=39 \[ -\frac {\log (1-a x)}{a^4}-\frac {x}{a^3}-\frac {x^2}{2 a^2}-\frac {x^3}{3 a} \]
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Rubi [A] time = 0.10, antiderivative size = 39, 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} \[ -\frac {x^2}{2 a^2}-\frac {x}{a^3}-\frac {\log (1-a x)}{a^4}-\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x^3}{1-a x} \, dx\\ &=\int \left (-\frac {1}{a^3}-\frac {x}{a^2}-\frac {x^2}{a}-\frac {1}{a^3 (-1+a x)}\right ) \, dx\\ &=-\frac {x}{a^3}-\frac {x^2}{2 a^2}-\frac {x^3}{3 a}-\frac {\log (1-a x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.00 \[ -\frac {\log (1-a x)}{a^4}-\frac {x}{a^3}-\frac {x^2}{2 a^2}-\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 34, normalized size = 0.87 \[ -\frac {2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log \left (a x - 1\right )}{6 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 36, normalized size = 0.92 \[ -\frac {2 \, a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{6 \, a^{3}} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 0.90 \[ -\frac {x^{3}}{3 a}-\frac {x^{2}}{2 a^{2}}-\frac {x}{a^{3}}-\frac {\ln \left (a x -1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 35, normalized size = 0.90 \[ -\frac {2 \, a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{6 \, a^{3}} - \frac {\log \left (a x - 1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.87 \[ -\frac {\ln \left (a\,x-1\right )}{a^4}-\frac {x}{a^3}-\frac {x^3}{3\,a}-\frac {x^2}{2\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 31, normalized size = 0.79 \[ - \frac {x^{3}}{3 a} - \frac {x^{2}}{2 a^{2}} - \frac {x}{a^{3}} - \frac {\log {\left (a x - 1 \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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