Optimal. Leaf size=33 \[ \frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3}+\frac {2 \log (1-a x)}{a^3} \]
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Rubi [A]
time = 0.03, 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 = {6302, 6261, 78}
\begin {gather*} \frac {2 \log (1-a x)}{a^3}+\frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6261
Rule 6302
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 = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {2 x}{a^2}+\frac {x^2}{a}+\frac {x^3}{3}+\frac {2 \log (1-a x)}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 34, normalized size = 1.03
method | result | size |
norman | \(\frac {x^{2}}{a}+\frac {x^{3}}{3}+\frac {2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(31\) |
risch | \(\frac {x^{2}}{a}+\frac {x^{3}}{3}+\frac {2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(31\) |
default | \(\frac {\frac {1}{3} a^{2} x^{3}+a \,x^{2}+2 x}{a^{2}}+\frac {2 \ln \left (a x -1\right )}{a^{3}}\) | \(34\) |
meijerg | \(-\frac {-\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{12}-\ln \left (-a x +1\right )}{a^{3}}+\frac {\frac {a x \left (3 a x +6\right )}{6}+\ln \left (-a x +1\right )}{a^{3}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 34, normalized size = 1.03 \begin {gather*} \frac {a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} + \frac {2 \, \log \left (a x - 1\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 33, normalized size = 1.00 \begin {gather*} \frac {a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log \left (a x - 1\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 27, normalized size = 0.82 \begin {gather*} \frac {x^{3}}{3} + \frac {x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 \log {\left (a x - 1 \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 38, normalized size = 1.15 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 30, normalized size = 0.91 \begin {gather*} \frac {2\,\ln \left (a\,x-1\right )}{a^3}+\frac {2\,x}{a^2}+\frac {x^3}{3}+\frac {x^2}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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