Optimal. Leaf size=46 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]
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Rubi [A] time = 0.0407638, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx &=\frac{\sqrt{1-x^2} \int \frac{1}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx}{\sqrt{-1+x^2}}\\ &=\frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{-1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0817452, size = 48, normalized size = 1.04 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{4 \sqrt{3}-7}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.106, size = 117, normalized size = 2.5 \begin{align*}{\frac{-i \left ( -2+\sqrt{3} \right ) }{ \left ( 4\,\sqrt{3}-7 \right ) \left ( -{x}^{4}+4\,\sqrt{3}{x}^{2}-6\,{x}^{2}-4\,\sqrt{3}+7 \right ) }{\it EllipticF} \left ({\frac{ix}{-2+\sqrt{3}}},2\,i-i\sqrt{3} \right ) \sqrt{-{x}^{2}+1}\sqrt{- \left ( -{x}^{2}+4\,\sqrt{3}-7 \right ) \left ( 7-4\,\sqrt{3} \right ) }\sqrt{{x}^{2}-1}\sqrt{7+{x}^{2}-4\,\sqrt{3}}} \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{x^{2} - 4 \, \sqrt{3} + 7} \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{{\left (x^{2} + 4 \, \sqrt{3} + 7\right )} \sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}{x^{6} + 13 \, x^{4} - 13 \, 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{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \sqrt{x^{2} - 4 \sqrt{3} + 7}}\, 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{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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