Optimal. Leaf size=33 \[ -\frac {2 \log (a x+1)}{a^3}+\frac {2 x}{a^2}-\frac {x^2}{a}+\frac {x^3}{3} \]
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Rubi [A] time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 6126, 77} \[ \frac {2 x}{a^2}-\frac {2 \log (a x+1)}{a^3}-\frac {x^2}{a}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} x^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} x^2 \, dx\\ &=-\int \frac {x^2 (1-a x)}{1+a x} \, dx\\ &=-\int \left (-\frac {2}{a^2}+\frac {2 x}{a}-x^2+\frac {2}{a^2 (1+a x)}\right ) \, dx\\ &=\frac {2 x}{a^2}-\frac {x^2}{a}+\frac {x^3}{3}-\frac {2 \log (1+a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ -\frac {2 \log (a x+1)}{a^3}+\frac {2 x}{a^2}-\frac {x^2}{a}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.00 \[ \frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 6 \, \log \left (a x + 1\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 38, normalized size = 1.15 \[ \frac {a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x}{3 \, a^{3}} - \frac {2 \, \log \left ({\left | a x + 1 \right |}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.97 \[ \frac {2 x}{a^{2}}-\frac {x^{2}}{a}+\frac {x^{3}}{3}-\frac {2 \ln \left (a x +1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 1.03 \[ \frac {a^{2} x^{3} - 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} - \frac {2 \, \log \left (a x + 1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 31, normalized size = 0.94 \[ \frac {2\,x}{a^2}-\frac {2\,\ln \left (a\,x+1\right )}{a^3}+\frac {x^3}{3}-\frac {x^2}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 27, normalized size = 0.82 \[ \frac {x^{3}}{3} - \frac {x^{2}}{a} + \frac {2 x}{a^{2}} - \frac {2 \log {\left (a x + 1 \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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