Optimal. Leaf size=61 \[ \frac{\sqrt{c+d x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{x}{2}\right ),1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
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Rubi [A] time = 0.0124724, antiderivative size = 61, 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 = {418} \[ \frac{\sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4+x^2} \sqrt{c+d x^2}} \, dx &=\frac{\sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{c \sqrt{4+x^2} \sqrt{\frac{c+d x^2}{c \left (4+x^2\right )}}}\\ \end{align*}
Mathematica [C] time = 0.0346564, size = 47, normalized size = 0.77 \[ -\frac{i \sqrt{\frac{c+d x^2}{c}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{2}\right ),\frac{4 d}{c}\right )}{\sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 53, normalized size = 0.9 \begin{align*}{\frac{1}{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{1}{2}\sqrt{{\frac{c}{d}}}} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{-{\frac{d}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c + d x^{2}} \sqrt{x^{2} + 4}}\, 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{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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