Optimal. Leaf size=13 \[ 2 \sqrt {(\sin (x)+1) \sec (x)} \]
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Rubi [A] time = 0.15, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4397, 4400, 2705, 2671} \[ 2 \sqrt {(\sin (x)+1) \sec (x)} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2705
Rule 4397
Rule 4400
Rubi steps
\begin {align*} \int \sec (x) \sqrt {\sec (x)+\tan (x)} \, dx &=\int \sec (x) \sqrt {\sec (x) (1+\sin (x))} \, dx\\ &=\frac {\sqrt {\sec (x) (1+\sin (x))} \int \sec ^{\frac {3}{2}}(x) \sqrt {1+\sin (x)} \, dx}{\sqrt {\sec (x)} \sqrt {1+\sin (x)}}\\ &=\frac {\left (\sqrt {\cos (x)} \sqrt {\sec (x) (1+\sin (x))}\right ) \int \frac {\sqrt {1+\sin (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx}{\sqrt {1+\sin (x)}}\\ &=2 \sqrt {\sec (x) (1+\sin (x))}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 37, normalized size = 2.85 \[ 2 \sqrt {\frac {\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 21, normalized size = 1.62 \[ 2 \, \sqrt {\frac {\cos \relax (x) + \sin \relax (x) + 1}{\cos \relax (x) - \sin \relax (x) + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 55, normalized size = 4.23 \[ -\frac {4 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )^{2} - \tan \left (\frac {1}{2} \, x\right ) - 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right )}{\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{2} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 10, normalized size = 0.77 \[ 2 \sqrt {\sec \relax (x )+\tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sec \relax (x) + \tan \relax (x)} \sec \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 14, normalized size = 1.08 \[ 2\,\sqrt {\frac {1}{\cos \relax (x)}}\,\sqrt {\sin \relax (x)+1} \]
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 \[ \int \sqrt {\tan {\relax (x )} + \sec {\relax (x )}} \sec {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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