Optimal. Leaf size=10 \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
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Rubi [A] time = 0.0093696, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {424} \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
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Rule 424
Rubi steps
\begin{align*} \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}\\ \end{align*}
Mathematica [A] time = 0.0071131, size = 10, normalized size = 1. \[ \frac{E\left (\left .\sin ^{-1}(c x)\right |-1\right )}{c} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.063, size = 15, normalized size = 1.5 \begin{align*}{\frac{{\it EllipticE} \left ( x{\it csgn} \left ( c \right ) c,i \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}}{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{c^{2} x^{2} + 1}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}\, 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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