Optimal. Leaf size=9 \[ \frac {\log \left (\tanh ^{-1}(a x)\right )}{a} \]
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Rubi [A]
time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6093}
\begin {gather*} \frac {\log \left (\tanh ^{-1}(a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 6093
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \, dx &=\frac {\log \left (\tanh ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 9, normalized size = 1.00 \begin {gather*} \frac {\log \left (\tanh ^{-1}(a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 10, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\ln \left (\arctanh \left (a x \right )\right )}{a}\) | \(10\) |
default | \(\frac {\ln \left (\arctanh \left (a x \right )\right )}{a}\) | \(10\) |
risch | \(\frac {\ln \left (\ln \left (a x +1\right )-\ln \left (-a x +1\right )\right )}{a}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.26, size = 21, normalized size = 2.33 \begin {gather*} \frac {\log \left (-\log \left (a x + 1\right ) + \log \left (-a x + 1\right )\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs.
\(2 (9) = 18\).
time = 0.35, size = 20, normalized size = 2.22 \begin {gather*} \frac {\log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.77, size = 7, normalized size = 0.78 \begin {gather*} \frac {\log {\left (\operatorname {atanh}{\left (a x \right )} \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.39, size = 21, normalized size = 2.33 \begin {gather*} \frac {\log \left ({\left | \log \left (-\frac {a x + 1}{a x - 1}\right ) \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.82, size = 9, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {atanh}\left (a\,x\right )\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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