Optimal. Leaf size=7 \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3190, 216} \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {4-x^2}} \, dx,x,\sin (x)\right )\\ &=\sin ^{-1}\left (\frac {\sin (x)}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 7, normalized size = 1.00 \[ \sin ^{-1}\left (\frac {\sin (x)}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 3.02, size = 53, normalized size = 7.57 \[ \frac {1}{2} \, \arctan \left (\frac {\sqrt {\cos \relax (x)^{2} + 3} {\left (\cos \relax (x)^{2} + 1\right )} \sin \relax (x) - 4 \, \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} - 3}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 5, normalized size = 0.71 \[ \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 6, normalized size = 0.86 \[ \arcsin \left (\frac {\sin \relax (x )}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 5, normalized size = 0.71 \[ \arcsin \left (\frac {1}{2} \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 5, normalized size = 0.71 \[ \mathrm {asin}\left (\frac {\sin \relax (x)}{2}\right ) \]
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 \[ \int \frac {\cos {\relax (x )}}{\sqrt {- \left (\sin {\relax (x )} - 2\right ) \left (\sin {\relax (x )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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