Optimal. Leaf size=13 \[ 2 \sqrt{(\sin (x)+1) \sec (x)} \]
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Rubi [A] time = 0.145602, antiderivative size = 13, normalized size of antiderivative = 1., 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 4397
Rule 4400
Rule 2705
Rule 2671
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*}
Mathematica [B] time = 0.0446157, 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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Maple [A] time = 0.058, size = 10, normalized size = 0.8 \begin{align*} 2\,\sqrt{\sec \left ( x \right ) +\tan \left ( x \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sec \left (x\right ) + \tan \left (x\right )} \sec \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03483, size = 72, normalized size = 5.54 \begin{align*} 2 \, \sqrt{\frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan{\left (x \right )} + \sec{\left (x \right )}} \sec{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29174, size = 74, normalized size = 5.69 \begin{align*} -\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 \left (x\right )\right )}{\frac{\sqrt{-\tan \left (\frac{1}{2} \, x\right )^{2} + 1} - 1}{\tan \left (\frac{1}{2} \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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