Optimal. Leaf size=42 \[ \frac{\tan (e+f x) \sqrt{c-c \sec (e+f x)}}{2 f (a \sec (e+f x)+a)^{3/2}} \]
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Rubi [A] time = 0.134305, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {3950} \[ \frac{\tan (e+f x) \sqrt{c-c \sec (e+f x)}}{2 f (a \sec (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3950
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx &=\frac{\sqrt{c-c \sec (e+f x)} \tan (e+f x)}{2 f (a+a \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.183704, size = 42, normalized size = 1. \[ \frac{\csc (e+f x) \sqrt{c-c \sec (e+f x)}}{a f \sqrt{a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.284, size = 73, normalized size = 1.7 \begin{align*}{\frac{\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}{2\,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 ) }}}\sqrt{{\frac{a \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 [A] time = 1.52174, size = 73, normalized size = 1.74 \begin{align*} \frac{\sqrt{c}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{2 \, \sqrt{-a} a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.466875, size = 186, normalized size = 4.43 \begin{align*} \frac{\sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\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*} \int \frac{\sqrt{- c \left (\sec{\left (e + f x \right )} - 1\right )} \sec{\left (e + f x \right )}}{\left (a \left (\sec{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.67175, size = 116, normalized size = 2.76 \begin{align*} -\frac{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} 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 )}{2 \, \sqrt{-a c} a f{\left | c \right |} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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