Optimal. Leaf size=13 \[ -\frac {1}{2 a \tanh ^{-1}(a x)^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 13, 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 = {6095}
\begin {gather*} -\frac {1}{2 a \tanh ^{-1}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6095
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \tanh ^{-1}(a x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{2 a \tanh ^{-1}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {1}{2 a \arctanh \left (a x \right )^{2}}\) | \(12\) |
default | \(-\frac {1}{2 a \arctanh \left (a x \right )^{2}}\) | \(12\) |
risch | \(-\frac {2}{a \left (-\ln \left (a x +1\right )+\ln \left (-a x +1\right )\right )^{2}}\) | \(24\) |
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. 42 vs.
\(2 (11) = 22\).
time = 0.26, size = 42, normalized size = 3.23 \begin {gather*} -\frac {2}{a \log \left (a x + 1\right )^{2} - 2 \, a \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + a \log \left (-a x + 1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 22, normalized size = 1.69 \begin {gather*} -\frac {2}{a \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.90, size = 12, normalized size = 0.92 \begin {gather*} - \frac {1}{2 a \operatorname {atanh}^{2}{\left (a x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 1.69 \begin {gather*} -\frac {2}{a \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 23, normalized size = 1.77 \begin {gather*} -\frac {2}{a\,{\left (\ln \left (a\,x+1\right )-\ln \left (1-a\,x\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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