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 = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 12
Rule 63
Rule 208
Rule 266
Rule 857
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*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2}}{1-a x}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 52, normalized size = 1.27 \[ \frac {a x + {\left (a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} - 1}{a x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 74, normalized size = 1.80 \[ -\frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {2 \, a}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 1.41 \[ -\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{\left (x -\frac {1}{a}\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{\sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 58, normalized size = 1.41 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{\sqrt {-a^2}\,\left (\frac {a}{\sqrt {-a^2}}+x\,\sqrt {-a^2}\right )}-\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \]
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 \[ - \int \frac {1}{a x^{2} \sqrt {- a^{2} x^{2} + 1} - x \sqrt {- a^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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