Optimal. Leaf size=27 \[ x+\frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6260, 45}
\begin {gather*} \frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6260
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \, dx &=\int \frac {(1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (1+\frac {4}{(-1+a x)^2}+\frac {4}{-1+a x}\right ) \, dx\\ &=x+\frac {4}{a (1-a x)}+\frac {4 \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 26, normalized size = 0.96 \begin {gather*} x-\frac {4}{a (-1+a x)}+\frac {4 \log (1-a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 26, normalized size = 0.96
method | result | size |
default | \(x -\frac {4}{a \left (a x -1\right )}+\frac {4 \ln \left (a x -1\right )}{a}\) | \(26\) |
risch | \(x -\frac {4}{a \left (a x -1\right )}+\frac {4 \ln \left (a x -1\right )}{a}\) | \(26\) |
norman | \(\frac {a^{2} x^{3}-4 a \,x^{2}-5 x}{a^{2} x^{2}-1}+\frac {4 \ln \left (a x -1\right )}{a}\) | \(42\) |
meijerg | \(\frac {\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right )}{5 a^{4} \left (-a^{2} x^{2}+1\right )}-\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{a^{5}}}{2 \sqrt {-a^{2}}}+\frac {\frac {2 a^{2} x^{2}}{-a^{2} x^{2}+1}+2 \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {3 \left (\frac {x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (-a^{2} x^{2}+1\right )}-\frac {\left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+\frac {2 a \,x^{2}}{-a^{2} x^{2}+1}+\frac {\frac {2 x \sqrt {-a^{2}}}{-2 a^{2} x^{2}+2}+\frac {\sqrt {-a^{2}}\, \arctanh \left (a x \right )}{a}}{2 \sqrt {-a^{2}}}\) | \(214\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 26, normalized size = 0.96 \begin {gather*} x + \frac {4 \, \log \left (a x - 1\right )}{a} - \frac {4}{a^{2} x - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 1.41 \begin {gather*} \frac {a^{2} x^{2} - a x + 4 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 4}{a^{2} x - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.70 \begin {gather*} x - \frac {4}{a^{2} x - a} + \frac {4 \log {\left (a x - 1 \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 0.96 \begin {gather*} x + \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {4}{{\left (a x - 1\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 25, normalized size = 0.93 \begin {gather*} x-\frac {4}{a\,\left (a\,x-1\right )}+\frac {4\,\ln \left (a\,x-1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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