Optimal. Leaf size=9 \[ \sin ^{-1}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3186, 216} \[ \sin ^{-1}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 3186
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\sqrt {1+\cos ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2}} \, dx,x,\sin (x)\right )\\ &=\sin ^{-1}\left (\frac {\sin (x)}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \sin ^{-1}\left (\frac {\sin (x)}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 49, normalized size = 5.44 \[ \frac {1}{2} \, \arctan \left (\frac {\sqrt {\cos \relax (x)^{2} + 1} \cos \relax (x)^{2} \sin \relax (x) - \cos \relax (x) \sin \relax (x)}{\cos \relax (x)^{4} + \cos \relax (x)^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 8, normalized size = 0.89 \[ \arcsin \left (\frac {1}{2} \, \sqrt {2} \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.88, size = 33, normalized size = 3.67 \[ -\frac {\sqrt {\left (1+\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}\, \arcsin \left (\cos ^{2}\relax (x )\right )}{2 \sin \relax (x ) \sqrt {1+\cos ^{2}\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 8, normalized size = 0.89 \[ \arcsin \left (\frac {1}{2} \, \sqrt {2} \sin \relax (x)\right ) \]
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 \[ \int \frac {\cos \relax (x)}{\sqrt {{\cos \relax (x)}^2+1}} \,d x \]
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 \[ \int \frac {\cos {\relax (x )}}{\sqrt {\cos ^{2}{\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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