Optimal. Leaf size=37 \[ \frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}-\frac {1}{a \sqrt {a-a x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5958} \[ \frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}-\frac {1}{a \sqrt {a-a x^2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx &=-\frac {1}{a \sqrt {a-a x^2}}+\frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 0.81 \[ \frac {\sqrt {a-a x^2} \left (1-x \tanh ^{-1}(x)\right )}{a^2 \left (x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 42, normalized size = 1.14 \[ -\frac {\sqrt {-a x^{2} + a} {\left (x \log \left (-\frac {x + 1}{x - 1}\right ) - 2\right )}}{2 \, {\left (a^{2} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 54, normalized size = 1.46 \[ -\frac {\sqrt {-a x^{2} + a} x \log \left (-\frac {x + 1}{x - 1}\right )}{2 \, {\left (a x^{2} - a\right )} a} - \frac {1}{\sqrt {-a x^{2} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 52, normalized size = 1.41 \[ -\frac {\left (\arctanh \relax (x )-1\right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{2 \left (-1+x \right ) a^{2}}-\frac {\left (1+\arctanh \relax (x )\right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{2 \left (1+x \right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 63, normalized size = 1.70 \[ \frac {x \operatorname {artanh}\relax (x)}{\sqrt {-a x^{2} + a} a} - \frac {\frac {\sqrt {-a x^{2} + a}}{a x + a} - \frac {\sqrt {-a x^{2} + a}}{a x - a}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {atanh}\relax (x)}{{\left (a-a\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\relax (x )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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