Optimal. Leaf size=20 \[ -\frac {1}{x}+a \log (x)-a \log (1-a x) \]
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Rubi [A]
time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6285, 46}
\begin {gather*} a \log (x)-a \log (1-a x)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \sqrt {1-a^2 x^2}} \, dx &=\int \frac {1}{x^2 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx\\ &=-\frac {1}{x}+a \log (x)-a \log (1-a x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+a \log (x)-a \log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 20, normalized size = 1.00
method | result | size |
default | \(-\frac {1}{x}+a \ln \left (x \right )-a \ln \left (a x -1\right )\) | \(20\) |
norman | \(-\frac {1}{x}+a \ln \left (x \right )-a \ln \left (a x -1\right )\) | \(20\) |
risch | \(-\frac {1}{x}+a \ln \left (-x \right )-a \ln \left (a x -1\right )\) | \(22\) |
meijerg | \(\frac {a \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2}-\frac {a^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 19, normalized size = 0.95 \begin {gather*} -a \log \left (a x - 1\right ) + 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 = 22, normalized size = 1.10 \begin {gather*} -\frac {a x \log \left (a x - 1\right ) - 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 = 15, normalized size = 0.75 \begin {gather*} - 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 = 0.41, size = 21, normalized size = 1.05 \begin {gather*} -a \log \left ({\left | a x - 1 \right |}\right ) + 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 = 0.87, size = 16, normalized size = 0.80 \begin {gather*} 2\,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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