Optimal. Leaf size=9 \[ \sin ^{-1}\left (\frac{\sin (x)}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0211736, antiderivative size = 9, normalized size of antiderivative = 1., 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 3186
Rule 216
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*}
Mathematica [A] time = 0.0073821, size = 9, normalized size = 1. \[ \sin ^{-1}\left (\frac{\sin (x)}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.377, size = 33, normalized size = 3.7 \begin{align*} -{\frac{\arcsin \left ( \left ( \cos \left ( x \right ) \right ) ^{2} \right ) }{2\,\sin \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1+ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42654, size = 11, normalized size = 1.22 \begin{align*} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.716, size = 162, normalized size = 18. \begin{align*} \frac{1}{2} \, \arctan \left (\frac{\sqrt{\cos \left (x\right )^{2} + 1} \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 1}\right ) + \frac{1}{2} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33444, size = 11, normalized size = 1.22 \begin{align*} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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