Optimal. Leaf size=41 \[ \frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.100431, antiderivative size = 41, 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} \[ \frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}} \]
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))^2} \, dx &=\frac{2 c \tan (e+f x)}{3 f (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.150477, size = 55, normalized size = 1.34 \[ -\frac{\cos ^2(e+f x) \csc \left (\frac{1}{2} (e+f x)\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}{6 a^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.256, size = 53, normalized size = 1.3 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3\,f{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\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 = 1.52617, size = 147, normalized size = 3.59 \begin{align*} -\frac{\sqrt{2} \sqrt{c} - \frac{2 \, \sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}{6 \, a^{2} 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 = 0.459774, size = 142, normalized size = 3.46 \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 )^{2}}{3 \,{\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \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 ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39376, size = 89, normalized size = 2.17 \begin{align*} \frac{\sqrt{2}{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} \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 )}{6 \, a^{2} c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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