Optimal. Leaf size=9 \[ \sin ^{-1}\left (\frac {\tan (x)}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 216
Rule 4146
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*}
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Mathematica [B] time = 0.04, 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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fricas [B] time = 0.83, size = 25, normalized size = 2.78 \[ -\arctan \left (\frac {\sqrt {\frac {4 \, \cos \relax (x)^{2} - 1}{\cos \relax (x)^{2}}} \cos \relax (x)}{\sin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \relax (x)^{2}}{\sqrt {-\sec \relax (x)^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 103, normalized size = 11.44 \[ -\frac {\sqrt {2}\, \sqrt {\frac {2 \cos \relax (x )-1}{1+\cos \relax (x )}}\, \sqrt {6}\, \sqrt {\frac {1+2 \cos \relax (x )}{1+\cos \relax (x )}}\, \left (\EllipticF \left (\frac {\sqrt {3}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{3}\right )-2 \EllipticPi \left (\frac {\sqrt {3}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{3}, \frac {1}{3}\right )\right ) \left (\sin ^{2}\relax (x )\right ) \sqrt {3}}{9 \sqrt {\frac {4 \left (\cos ^{2}\relax (x )\right )-1}{\cos \relax (x )^{2}}}\, \cos \relax (x ) \left (-1+\cos \relax (x )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 8, normalized size = 0.89 \[ \arcsin \left (\frac {1}{3} \, \sqrt {3} \tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {1}{{\cos \relax (x)}^2\,\sqrt {4-\frac {1}{{\cos \relax (x)}^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{\sqrt {- \left (\sec {\relax (x )} - 2\right ) \left (\sec {\relax (x )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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