Optimal. Leaf size=13 \[ 2 \text{EllipticF}\left (\sin ^{-1}(x),-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.0137832, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {423, 424, 248, 221} \[ 2 F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 423
Rule 424
Rule 248
Rule 221
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \, dx &=2 \int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2}} \, dx-\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx\\ &=-E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 \int \frac{1}{\sqrt{1-x^4}} \, dx\\ &=-E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0031746, size = 12, normalized size = 0.92 \[ -i E\left (\left .i \sinh ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 14, normalized size = 1.1 \begin{align*} -{\it EllipticE} \left ( x,i \right ) +2\,{\it EllipticF} \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\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{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{x^{2} + 1}}\, dx \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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