Optimal. Leaf size=13 \[ \frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \sin (3 x)\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3190, 216} \[ \frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \sin (3 x)\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (3 x)}{\sqrt {4-\sin ^2(3 x)}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-x^2}} \, dx,x,\sin (3 x)\right )\\ &=\frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \sin (3 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 13, normalized size = 1.00 \[ \frac {1}{3} \sin ^{-1}\left (\frac {1}{2} \sin (3 x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 71, normalized size = 5.46 \[ \frac {1}{6} \, \arctan \left (\frac {\sqrt {\cos \left (3 \, x\right )^{2} + 3} {\left (\cos \left (3 \, x\right )^{2} + 1\right )} \sin \left (3 \, x\right ) - 4 \, \cos \left (3 \, x\right ) \sin \left (3 \, x\right )}{\cos \left (3 \, x\right )^{4} + 6 \, \cos \left (3 \, x\right )^{2} - 3}\right ) + \frac {1}{6} \, \arctan \left (\frac {\sin \left (3 \, x\right )}{\cos \left (3 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \sin \left (3 \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 10, normalized size = 0.77 \[ \frac {\arcsin \left (\frac {\sin \left (3 x \right )}{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 9, normalized size = 0.69 \[ \frac {1}{3} \, \arcsin \left (\frac {1}{2} \, \sin \left (3 \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 9, normalized size = 0.69 \[ \frac {\mathrm {asin}\left (\frac {\sin \left (3\,x\right )}{2}\right )}{3} \]
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 \[ \int \frac {\cos {\left (3 x \right )}}{\sqrt {- \left (\sin {\left (3 x \right )} - 2\right ) \left (\sin {\left (3 x \right )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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