Optimal. Leaf size=39 \[ \frac {4 x}{a}+\frac {x^2}{2}+\frac {4}{a^2 (1-a x)}+\frac {8 \log (1-a x)}{a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6261, 78}
\begin {gather*} \frac {4}{a^2 (1-a x)}+\frac {8 \log (1-a x)}{a^2}+\frac {4 x}{a}+\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6261
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} x \, dx &=\int \frac {x (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac {4}{a}+x+\frac {4}{a (-1+a x)^2}+\frac {8}{a (-1+a x)}\right ) \, dx\\ &=\frac {4 x}{a}+\frac {x^2}{2}+\frac {4}{a^2 (1-a x)}+\frac {8 \log (1-a x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 39, normalized size = 1.00 \begin {gather*} \frac {4 x}{a}+\frac {x^2}{2}+\frac {4}{a^2 (1-a x)}+\frac {8 \log (1-a x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 39, normalized size = 1.00
method | result | size |
risch | \(\frac {x^{2}}{2}+\frac {4 x}{a}-\frac {4}{\left (a x -1\right ) a^{2}}+\frac {8 \ln \left (a x -1\right )}{a^{2}}\) | \(36\) |
default | \(\frac {\frac {1}{2} a \,x^{2}+4 x}{a}-\frac {4}{\left (a x -1\right ) a^{2}}+\frac {8 \ln \left (a x -1\right )}{a^{2}}\) | \(39\) |
norman | \(\frac {-\frac {9 x^{2}}{2}+\frac {a^{2} x^{4}}{2}-\frac {8 x}{a}+4 x^{3} a}{a^{2} x^{2}-1}+\frac {8 \ln \left (a x -1\right )}{a^{2}}\) | \(51\) |
meijerg | \(-\frac {-\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )}-2 \ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}+\frac {\frac {2 x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {6 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}}{a \sqrt {-a^{2}}}+\frac {\frac {3 a^{2} x^{2}}{-a^{2} x^{2}+1}+3 \ln \left (-a^{2} x^{2}+1\right )}{a^{2}}-\frac {2 \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{a \sqrt {-a^{2}}}+\frac {x^{2}}{-2 a^{2} x^{2}+2}\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 41, normalized size = 1.05 \begin {gather*} \frac {a x^{2} + 8 \, x}{2 \, a} - \frac {4}{a^{3} x - a^{2}} + \frac {8 \, \log \left (a x - 1\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 49, normalized size = 1.26 \begin {gather*} \frac {a^{3} x^{3} + 7 \, a^{2} x^{2} - 8 \, a x + 16 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 8}{2 \, {\left (a^{3} x - a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 31, normalized size = 0.79 \begin {gather*} \frac {x^{2}}{2} - \frac {4}{a^{3} x - a^{2}} + \frac {4 x}{a} + \frac {8 \log {\left (a x - 1 \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 44, normalized size = 1.13 \begin {gather*} \frac {8 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{2}} + \frac {a^{4} x^{2} + 8 \, a^{3} x}{2 \, a^{4}} - \frac {4}{{\left (a x - 1\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 38, normalized size = 0.97 \begin {gather*} \frac {8\,\ln \left (a\,x-1\right )}{a^2}+\frac {4\,x}{a}+\frac {x^2}{2}+\frac {4}{a\,\left (a-a^2\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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