Optimal. Leaf size=21 \[ -\frac {1}{x}+2 a \log (x)-2 a \log (1-a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6261, 78}
\begin {gather*} 2 a \log (x)-2 a \log (1-a x)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {1+a x}{x^2 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 a}{x}-\frac {2 a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+2 a \log (x)-2 a \log (1-a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+2 a \log (x)-2 a \log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 21, normalized size = 1.00
method | result | size |
default | \(-\frac {1}{x}+2 a \ln \left (x \right )-2 \ln \left (a x -1\right ) a\) | \(21\) |
norman | \(-\frac {1}{x}+2 a \ln \left (x \right )-2 \ln \left (a x -1\right ) a\) | \(21\) |
risch | \(-\frac {1}{x}-2 \ln \left (a x -1\right ) a +2 \ln \left (-x \right ) a\) | \(23\) |
meijerg | \(a \arctanh \left (a x \right )+a \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )-\frac {a^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 0.95 \begin {gather*} -2 \, a \log \left (a x - 1\right ) + 2 \, a \log \left (x\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 23, normalized size = 1.10 \begin {gather*} -\frac {2 \, a x \log \left (a x - 1\right ) - 2 \, a x \log \left (x\right ) + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 17, normalized size = 0.81 \begin {gather*} - 2 a \left (- \log {\left (x \right )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 22, normalized size = 1.05 \begin {gather*} -2 \, a \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 16, normalized size = 0.76 \begin {gather*} 4\,a\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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