Optimal. Leaf size=90 \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.0792337, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {857, 835, 807, 266, 63, 208} \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 857
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{\int \frac{-3 a^2-2 a^3 x}{x^3 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{\int \frac{4 a^3+3 a^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{1}{2} \left (3 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{1}{4} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0588669, size = 63, normalized size = 0.7 \[ \frac{1}{2} \left (\frac{\left (-4 a^2 x^2+a x+1\right ) \sqrt{1-a^2 x^2}}{x^2 (a x-1)}-3 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 94, normalized size = 1. \begin{align*} -{\frac{3\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{a\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{a}{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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63976, size = 192, normalized size = 2.13 \begin{align*} \frac{2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (4 \, a^{2} x^{2} - a x - 1\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \,{\left (a x^{3} - x^{2}\right )}} \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^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \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 = 1.26761, size = 288, normalized size = 3.2 \begin{align*} -\frac{{\left (a^{3} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{3 \, a^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} - \frac{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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