Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right ) \]
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Rubi [A] time = 0.0024596, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-a^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-a^2+x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{x}{\sqrt{-a^2+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{-a^2+x^2}}\right )\\ \end{align*}
Mathematica [B] time = 0.0027682, size = 46, normalized size = 2.88 \[ \frac{1}{2} \log \left (\frac{x}{\sqrt{x^2-a^2}}+1\right )-\frac{1}{2} \log \left (1-\frac{x}{\sqrt{x^2-a^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 15, normalized size = 0.9 \begin{align*} \ln \left ( x+\sqrt{-{a}^{2}+{x}^{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974328, size = 24, normalized size = 1.5 \begin{align*} \log \left (2 \, x + 2 \, \sqrt{-a^{2} + x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88426, size = 39, normalized size = 2.44 \begin{align*} -\log \left (-x + \sqrt{-a^{2} + x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.0204, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \operatorname{acosh}{\left (\frac{x}{a} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\- i \operatorname{asin}{\left (\frac{x}{a} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05721, size = 26, normalized size = 1.62 \begin{align*} -\log \left ({\left | -x + \sqrt{-a^{2} + x^{2}} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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