Optimal. Leaf size=64 \[ -\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.0534527, antiderivative size = 64, 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, 807, 266, 63, 208} \[ -\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 857
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 (1-a x) \sqrt{1-a^2 x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-\frac{\int \frac{-2 a^2-a^3 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}+a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (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}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x (1-a x)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0445593, size = 50, normalized size = 0.78 \[ \frac{(1-2 a x) \sqrt{1-a^2 x^2}}{x (a x-1)}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 73, normalized size = 1.1 \begin{align*} -a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -{\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55587, size = 151, normalized size = 2.36 \begin{align*} \frac{a^{2} x^{2} - a x +{\left (a^{2} x^{2} - a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - 1\right )}}{a x^{2} - x} \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^{3} \sqrt{- a^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31984, size = 203, normalized size = 3.17 \begin{align*} -\frac{a^{2} \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{{\left (a^{2} - \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, x{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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