Optimal. Leaf size=32 \[ \frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.0469097, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5994, 216} \[ \frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 216
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a}\\ &=\frac{\sin ^{-1}(a x)}{a^2}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.031359, size = 29, normalized size = 0.91 \[ \frac{\sin ^{-1}(a x)-\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.222, size = 81, normalized size = 2.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) }{{a}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{i}{{a}^{2}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+i \right ) }-{\frac{i}{{a}^{2}}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-i \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44362, size = 57, normalized size = 1.78 \begin{align*} \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a} - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07488, size = 135, normalized size = 4.22 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 4 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{atanh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21385, size = 63, normalized size = 1.97 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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