Optimal. Leaf size=43 \[ \frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {x}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5994, 191} \[ \frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {x}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac {x}{a \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 27, normalized size = 0.63 \[ \frac {\tanh ^{-1}(a x)-a x}{a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 51, normalized size = 1.19 \[ \frac {\sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{2 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 61, normalized size = 1.42 \[ \frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} a} + \frac {\log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 66, normalized size = 1.53 \[ -\frac {\left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 39, normalized size = 0.91 \[ -\frac {x}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {\operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]
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 \[ \int \frac {x\,\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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