Optimal. Leaf size=39 \[ \frac{\sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
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Rubi [A] time = 0.0200224, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {421, 419} \[ \frac{\sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx &=\frac{\sqrt{1+\frac{d x^2}{c}} \int \frac{1}{\sqrt{4-x^2} \sqrt{1+\frac{d x^2}{c}}} \, dx}{\sqrt{c+d x^2}}\\ &=\frac{\sqrt{1+\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0379465, size = 40, normalized size = 1.03 \[ \frac{\sqrt{\frac{c+d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 38, normalized size = 1. \begin{align*}{\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ){\frac{1}{\sqrt{d{x}^{2}+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{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{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{d x^{2} + c} \sqrt{-x^{2} + 4}}{d x^{4} +{\left (c - 4 \, d\right )} x^{2} - 4 \, c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.21516, size = 20, normalized size = 0.51 \begin{align*} \begin{cases} \frac{F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{\sqrt{c}} & \text{for}\: x > -2 \wedge x < 2 \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{1}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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