Optimal. Leaf size=9 \[ \sinh ^{-1}\left (\frac {\sin (x)}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 9, 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 = {3186, 215} \[ \sinh ^{-1}\left (\frac {\sin (x)}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 3186
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\sqrt {4-\cos ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+x^2}} \, dx,x,\sin (x)\right )\\ &=\sinh ^{-1}\left (\frac {\sin (x)}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \sinh ^{-1}\left (\frac {\sin (x)}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.21, size = 39, normalized size = 4.33 \[ \frac {1}{4} \, \log \left (8 \, \cos \relax (x)^{4} - 4 \, {\left (2 \, \cos \relax (x)^{2} - 5\right )} \sqrt {-\cos \relax (x)^{2} + 4} \sin \relax (x) - 40 \, \cos \relax (x)^{2} + 41\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 16, normalized size = 1.78 \[ -\log \left (\sqrt {\sin \relax (x)^{2} + 3} - \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.15, size = 53, normalized size = 5.89 \[ -\frac {\sqrt {-\left (-4+\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}\, \ln \left (-\left (\sin ^{2}\relax (x )\right )+\sqrt {\sin ^{4}\relax (x )+3 \left (\sin ^{2}\relax (x )\right )}-\frac {3}{2}\right )}{2 \sin \relax (x ) \sqrt {4-\left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.74, size = 8, normalized size = 0.89 \[ \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} \sin \relax (x)\right ) \]
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 \[ \int \frac {\cos \relax (x)}{\sqrt {4-{\cos \relax (x)}^2}} \,d x \]
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 \[ \int \frac {\cos {\relax (x )}}{\sqrt {- \left (\cos {\relax (x )} - 2\right ) \left (\cos {\relax (x )} + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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