Optimal. Leaf size=41 \[ \frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.0397202, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {857, 12, 266, 63, 208} \[ \frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 857
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (1-a x) \sqrt{1-a^2 x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2}}{1-a x}+\frac{\int \frac{a^2}{x \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{1-a x}+\int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{\sqrt{1-a^2 x^2}}{1-a x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1-a^2 x^2}}{1-a x}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a^2}\\ &=\frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0301259, size = 41, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 58, normalized size = 1.4 \begin{align*} -{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\frac{1}{a}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{\sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58115, size = 116, normalized size = 2.83 \begin{align*} \frac{a x +{\left (a x - 1\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1} - 1}{a x - 1} \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{1}{a x^{2} \sqrt{- a^{2} x^{2} + 1} - x \sqrt{- a^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29234, size = 100, normalized size = 2.44 \begin{align*} -\frac{a \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{{\left | a \right |}} + \frac{2 \, a}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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