Optimal. Leaf size=4 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.0052178, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {424} \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx &=E\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [A] time = 0.003282, size = 4, normalized size = 1. \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 5, normalized size = 1.3 \begin{align*}{\it EllipticE} \left ( x,i \right ) \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{\sqrt{x^{2} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} + 1} \sqrt{-x^{2} + 1}}{x^{2} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.13838, size = 10, normalized size = 2.5 \begin{align*} \begin{cases} E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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