Optimal. Leaf size=13 \[ -\frac {1}{2 a \tanh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.03, 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 = {5948} \[ -\frac {1}{2 a \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 5948
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 \[ -\frac {1}{2 a \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 22, normalized size = 1.69 \[ -\frac {2}{a \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 22, normalized size = 1.69 \[ -\frac {2}{a \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 0.92 \[ -\frac {1}{2 a \arctanh \left (a x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 42, normalized size = 3.23 \[ -\frac {2}{a \log \left (a x + 1\right )^{2} - 2 \, a \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + a \log \left (-a x + 1\right )^{2}} \]
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 \[ -\frac {2}{a\,{\left (\ln \left (a\,x+1\right )-\ln \left (1-a\,x\right )\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 12, normalized size = 0.92 \[ - \frac {1}{2 a \operatorname {atanh}^{2}{\left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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