Optimal. Leaf size=47 \[ \frac {4}{a^3 (1-a x)}+\frac {12 \log (1-a x)}{a^3}+\frac {8 x}{a^2}+\frac {2 x^2}{a}+\frac {x^3}{3} \]
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Rubi [A] time = 0.06, antiderivative size = 47, 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, 88} \[ \frac {8 x}{a^2}+\frac {4}{a^3 (1-a x)}+\frac {12 \log (1-a x)}{a^3}+\frac {2 x^2}{a}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} x^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^2 \, dx\\ &=\int \frac {x^2 (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac {8}{a^2}+\frac {4 x}{a}+x^2+\frac {4}{a^2 (-1+a x)^2}+\frac {12}{a^2 (-1+a x)}\right ) \, dx\\ &=\frac {8 x}{a^2}+\frac {2 x^2}{a}+\frac {x^3}{3}+\frac {4}{a^3 (1-a x)}+\frac {12 \log (1-a x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 1.00 \[ \frac {4}{a^3 (1-a x)}+\frac {12 \log (1-a x)}{a^3}+\frac {8 x}{a^2}+\frac {2 x^2}{a}+\frac {x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 57, normalized size = 1.21 \[ \frac {a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 24 \, a x + 36 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 12}{3 \, {\left (a^{4} x - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 69, normalized size = 1.47 \[ \frac {{\left (a x - 1\right )}^{3} {\left (\frac {9}{a x - 1} + \frac {39}{{\left (a x - 1\right )}^{2}} + 1\right )}}{3 \, a^{3}} - \frac {12 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a^{3}} - \frac {4}{{\left (a x - 1\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.94 \[ \frac {x^{3}}{3}+\frac {2 x^{2}}{a}+\frac {8 x}{a^{2}}+\frac {12 \ln \left (a x -1\right )}{a^{3}}-\frac {4}{a^{3} \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 1.04 \[ -\frac {4}{a^{4} x - a^{3}} + \frac {a^{2} x^{3} + 6 \, a x^{2} + 24 \, x}{3 \, a^{2}} + \frac {12 \, \log \left (a x - 1\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 49, normalized size = 1.04 \[ \frac {12\,\ln \left (a\,x-1\right )}{a^3}-\frac {4}{a\,\left (a^3\,x-a^2\right )}+\frac {8\,x}{a^2}+\frac {x^3}{3}+\frac {2\,x^2}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 39, normalized size = 0.83 \[ \frac {x^{3}}{3} - \frac {4}{a^{4} x - a^{3}} + \frac {2 x^{2}}{a} + \frac {8 x}{a^{2}} + \frac {12 \log {\left (a x - 1 \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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