Optimal. Leaf size=40 \[ \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 40, 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 = {5958} \[ \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 27, normalized size = 0.68 \[ \frac {a x \tanh ^{-1}(a x)-1}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 47, normalized size = 1.18 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2\right )}}{2 \, {\left (a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 59, normalized size = 1.48 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{2} x^{2} - 1\right )}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 38, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a x \arctanh \left (a x \right )-1\right )}{a \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 36, normalized size = 0.90 \[ \frac {x \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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