Optimal. Leaf size=23 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}(c x),-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
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Rubi [A] time = 0.0260898, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {423, 424, 248, 221} \[ \frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
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Rule 423
Rule 424
Rule 248
Rule 221
Rubi steps
\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{\sqrt{1+c^2 x^2}} \, dx &=2 \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}} \, dx-\int \frac{\sqrt{1+c^2 x^2}}{\sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}+2 \int \frac{1}{\sqrt{1-c^4 x^4}} \, dx\\ &=-\frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}+\frac{2 F\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0075945, size = 24, normalized size = 1.04 \[ \frac{E\left (\left .\sin ^{-1}\left (\sqrt{-c^2} x\right )\right |-1\right )}{\sqrt{-c^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.049, size = 28, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2\,{\it EllipticF} \left ( x{\it csgn} \left ( c \right ) c,i \right ) -{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \right ) \right ){\it csgn} \left ( c \right ) }{c}} \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{-c^{2} x^{2} + 1}}{\sqrt{c^{2} 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{-c^{2} x^{2} + 1}}{\sqrt{c^{2} 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 (c x - 1\right ) \left (c x + 1\right )}}{\sqrt{c^{2} 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{-c^{2} x^{2} + 1}}{\sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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