Optimal. Leaf size=43 \[ \frac {2 \log (1-a x)}{a^4}+\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4} \]
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Rubi [A] time = 0.05, antiderivative size = 43, 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} \[ \frac {x^2}{a^2}+\frac {2 x}{a^3}+\frac {2 \log (1-a x)}{a^4}+\frac {2 x^3}{3 a}+\frac {x^4}{4} \]
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^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^3 \, dx\\ &=-\int \frac {x^3 (1+a x)}{1-a x} \, dx\\ &=-\int \left (-\frac {2}{a^3}-\frac {2 x}{a^2}-\frac {2 x^2}{a}-x^3-\frac {2}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4}+\frac {2 \log (1-a x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.00 \[ \frac {2 \log (1-a x)}{a^4}+\frac {2 x}{a^3}+\frac {x^2}{a^2}+\frac {2 x^3}{3 a}+\frac {x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 42, normalized size = 0.98 \[ \frac {3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x + 24 \, \log \left (a x - 1\right )}{12 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 47, normalized size = 1.09 \[ \frac {3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x}{12 \, a^{4}} + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.91 \[ \frac {x^{4}}{4}+\frac {2 x^{3}}{3 a}+\frac {x^{2}}{a^{2}}+\frac {2 x}{a^{3}}+\frac {2 \ln \left (a x -1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 43, normalized size = 1.00 \[ \frac {3 \, a^{3} x^{4} + 8 \, a^{2} x^{3} + 12 \, a x^{2} + 24 \, x}{12 \, a^{3}} + \frac {2 \, \log \left (a x - 1\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.88 \[ \frac {2\,\ln \left (a\,x-1\right )}{a^4}+\frac {2\,x}{a^3}+\frac {x^4}{4}+\frac {2\,x^3}{3\,a}+\frac {x^2}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 37, normalized size = 0.86 \[ \frac {x^{4}}{4} + \frac {2 x^{3}}{3 a} + \frac {x^{2}}{a^{2}} + \frac {2 x}{a^{3}} + \frac {2 \log {\left (a x - 1 \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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