Optimal. Leaf size=42 \[ -\frac{\tan (e+f x) \sqrt{a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.139167, 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} \[ -\frac{\tan (e+f x) \sqrt{a \sec (e+f x)+a}}{2 f (c-c \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3950
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac{\sqrt{a+a \sec (e+f x)} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.251898, size = 62, normalized size = 1.48 \[ \frac{\tan \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)}}{c f (\sec (e+f x)-1) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.303, size = 60, normalized size = 1.4 \begin{align*} -{\frac{\sin \left ( fx+e \right ) }{2\,f\cos \left ( fx+e \right ) }\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.83427, size = 694, normalized size = 16.52 \begin{align*} -\frac{2 \,{\left ({\left (\sin \left (3 \, f x + 3 \, e\right ) + \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) -{\left (\cos \left (3 \, f x + 3 \, e\right ) + \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) +{\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - 2 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 2 \, \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 2 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + \sin \left (f x + e\right )\right )} \sqrt{a} \sqrt{c}}{{\left (c^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \cos \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, c^{2} \cos \left (f x + e\right )^{2} + c^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, c^{2} \sin \left (3 \, f x + 3 \, e\right )^{2} + 4 \, c^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 8 \, c^{2} \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, c^{2} \sin \left (f x + e\right )^{2} - 4 \, c^{2} \cos \left (f x + e\right ) + c^{2} - 2 \,{\left (2 \, c^{2} \cos \left (3 \, f x + 3 \, e\right ) - 2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) + 2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 \,{\left (2 \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, c^{2} \cos \left (f x + e\right ) + c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right ) - 4 \,{\left (2 \, c^{2} \cos \left (f x + e\right ) - c^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \,{\left (c^{2} \sin \left (3 \, f x + 3 \, e\right ) - c^{2} \sin \left (2 \, f x + 2 \, e\right ) + c^{2} \sin \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 8 \,{\left (c^{2} \sin \left (2 \, f x + 2 \, e\right ) - c^{2} \sin \left (f x + e\right )\right )} \sin \left (3 \, f x + 3 \, e\right )\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.461, 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 (c^{2} f \cos \left (f x + e\right ) - c^{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{a \left (\sec{\left (e + f x \right )} + 1\right )} \sec{\left (e + f x \right )}}{\left (- c \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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