Optimal. Leaf size=12 \[ -\frac{1}{\sqrt{\sec (2 x)+1}} \]
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Rubi [A] time = 0.0545603, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4339, 261} \[ -\frac{1}{\sqrt{\sec (2 x)+1}} \]
Antiderivative was successfully verified.
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Rule 4339
Rule 261
Rubi steps
\begin{align*} \int \frac{\sec (2 x) \tan (2 x)}{(1+\sec (2 x))^{3/2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{1}{x}\right )^{3/2} x^2} \, dx,x,\cos (2 x)\right )\right )\\ &=-\frac{1}{\sqrt{1+\sec (2 x)}}\\ \end{align*}
Mathematica [A] time = 0.0767221, size = 20, normalized size = 1.67 \[ -\frac{2 \cos ^2(x) \sec (2 x)}{(\sec (2 x)+1)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{\sqrt{1+\sec \left ( 2\,x \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974531, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\sqrt{\sec \left (2 \, x\right ) + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39472, size = 76, normalized size = 6.33 \begin{align*} -\frac{\sqrt{\frac{\cos \left (2 \, x\right ) + 1}{\cos \left (2 \, x\right )}} \cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.28341, size = 12, normalized size = 1. \begin{align*} - \frac{1}{\sqrt{\sec{\left (2 x \right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46652, size = 32, normalized size = 2.67 \begin{align*} \frac{\sqrt{2} \sqrt{-\tan \left (x\right )^{2} + 1}}{2 \, \mathrm{sgn}\left (\tan \left (x\right )^{2} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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