Optimal. Leaf size=41 \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {\cos (x)} \sqrt {a \cos (x)+a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2782, 205} \[ \frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {\cos (x)} \sqrt {a \cos (x)+a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2782
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos (x)} \sqrt {a+a \cos (x)}} \, dx &=-\left ((2 a) \operatorname {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (x)}{\sqrt {\cos (x)} \sqrt {a+a \cos (x)}}\right )\right )\\ &=\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {\cos (x)} \sqrt {a+a \cos (x)}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.78 \[ \frac {2 \cos \left (\frac {x}{2}\right ) \tan ^{-1}\left (\frac {\sin \left (\frac {x}{2}\right )}{\sqrt {\cos (x)}}\right )}{\sqrt {a (\cos (x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 105, normalized size = 2.56 \[ \left [\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a \cos \relax (x) + a} \sqrt {-\frac {1}{a}} \sqrt {\cos \relax (x)} \sin \relax (x) - 3 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) + 1}{\cos \relax (x)^{2} + 2 \, \cos \relax (x) + 1}\right ), \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \relax (x) + a} \sqrt {\cos \relax (x)} \sin \relax (x)}{2 \, {\left (\cos \relax (x)^{2} + \cos \relax (x)\right )} \sqrt {a}}\right )}{\sqrt {a}}\right ] \]
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 \[ \int \frac {1}{\sqrt {a \cos \relax (x) + a} \sqrt {\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 42, normalized size = 1.02 \[ -\frac {\sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \sqrt {a \left (\cos \relax (x )+1\right )}\, \arcsin \left (\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right ) \sqrt {2}}{\sqrt {\cos \relax (x )}\, a} \]
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 = -0.02 \[ \int \frac {1}{\sqrt {\cos \relax (x)}\,\sqrt {a+a\,\cos \relax (x)}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \left (\cos {\relax (x )} + 1\right )} \sqrt {\cos {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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