Optimal. Leaf size=39 \[ \frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.106548, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3953} \[ \frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3953
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx &=\frac{2 c \tan (e+f x)}{f (a+a \sec (e+f x)) \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.131101, size = 29, normalized size = 0.74 \[ -\frac{2 \cot (e+f x) \sqrt{c-c \sec (e+f x)}}{a f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.235, size = 43, normalized size = 1.1 \begin{align*} -2\,{\frac{\cos \left ( fx+e \right ) }{fa\sin \left ( fx+e \right ) }\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52542, size = 113, normalized size = 2.9 \begin{align*} -\frac{\sqrt{2} \sqrt{c} - \frac{\sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a f \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.456224, size = 103, normalized size = 2.64 \begin{align*} -\frac{2 \, \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \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*} \frac{\int \frac{\sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )}}{\sec{\left (e + f x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42032, size = 85, normalized size = 2.18 \begin{align*} -\frac{\sqrt{2} \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right ) \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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