Optimal. Leaf size=16 \[ \sqrt {2} \sin ^{-1}\left (\frac {\sin (x)}{\cos (x)+1}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 16, 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 = {2781, 216} \[ \sqrt {2} \sin ^{-1}\left (\frac {\sin (x)}{\cos (x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 2781
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos (x)} \sqrt {1+\cos (x)}} \, dx &=-\left (\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,-\frac {\sin (x)}{1+\cos (x)}\right )\right )\\ &=\sqrt {2} \sin ^{-1}\left (\frac {\sin (x)}{1+\cos (x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.88 \[ \frac {2 \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (\frac {\sin \left (\frac {x}{2}\right )}{\sqrt {\cos (x)}}\right )}{\sqrt {\cos (x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 31, normalized size = 1.94 \[ \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\cos \relax (x) + 1} \sqrt {\cos \relax (x)} \sin \relax (x)}{2 \, {\left (\cos \relax (x)^{2} + \cos \relax (x)\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\cos \relax (x) + 1} \sqrt {\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 36, normalized size = 2.25 \[ -\frac {\sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \sqrt {2+2 \cos \relax (x )}\, \arcsin \left (\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )}{\sqrt {\cos \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {\cos \relax (x)}\,\sqrt {\cos \relax (x)+1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\cos {\relax (x )} + 1} \sqrt {\cos {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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