Optimal. Leaf size=9 \[ \sin ^{-1}\left (\frac{\tan (x)}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0467953, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4146, 216} \[ \sin ^{-1}\left (\frac{\tan (x)}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 4146
Rule 216
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{\sqrt{4-\sec ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{3-x^2}} \, dx,x,\tan (x)\right )\\ &=\sin ^{-1}\left (\frac{\tan (x)}{\sqrt{3}}\right )\\ \end{align*}
Mathematica [B] time = 0.0437544, size = 43, normalized size = 4.78 \[ \frac{\sqrt{2 \cos (2 x)+1} \sec (x) \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{3-4 \sin ^2(x)}}\right )}{\sqrt{4-\sec ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.164, size = 103, normalized size = 11.4 \begin{align*} -{\frac{\sqrt{3}\sqrt{2}\sqrt{6} \left ( \sin \left ( x \right ) \right ) ^{2}}{9\,\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) }\sqrt{{\frac{2\,\cos \left ( x \right ) -1}{1+\cos \left ( x \right ) }}}\sqrt{{\frac{1+2\,\cos \left ( x \right ) }{1+\cos \left ( x \right ) }}} \left ({\it EllipticF} \left ({\frac{\sqrt{3} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},{\frac{1}{3}} \right ) -2\,{\it EllipticPi} \left ({\frac{\sqrt{3} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},1/3,1/3 \right ) \right ){\frac{1}{\sqrt{{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}-1}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41891, size = 11, normalized size = 1.22 \begin{align*} \arcsin \left (\frac{1}{3} \, \sqrt{3} \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11589, size = 76, normalized size = 8.44 \begin{align*} -\arctan \left (\frac{\sqrt{\frac{4 \, \cos \left (x\right )^{2} - 1}{\cos \left (x\right )^{2}}} \cos \left (x\right )}{\sin \left (x\right )}\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{\sec ^{2}{\left (x \right )}}{\sqrt{- \left (\sec{\left (x \right )} - 2\right ) \left (\sec{\left (x \right )} + 2\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{\sec \left (x\right )^{2}}{\sqrt{-\sec \left (x\right )^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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