Optimal. Leaf size=95 \[ \frac{\tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{2 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.278977, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3960, 3959, 3770} \[ \frac{\tan (e+f x)}{2 f (a \sec (e+f x)+a)^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x) \tanh ^{-1}(\cos (e+f x))}{2 a f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3960
Rule 3959
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}} \, dx &=\frac{\tan (e+f x)}{2 f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx}{2 a}\\ &=\frac{\tan (e+f x)}{2 f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x) \int \csc (e+f x) \, dx}{2 a \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{\tan (e+f x)}{2 f (a+a \sec (e+f x))^{3/2} \sqrt{c-c \sec (e+f x)}}-\frac{\tanh ^{-1}(\cos (e+f x)) \tan (e+f x)}{2 a f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.40295, size = 157, normalized size = 1.65 \[ -\frac{\sin \left (\frac{1}{2} (e+f x)\right ) \sec ^{\frac{3}{2}}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )+i \sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (1+2 (\cos (e+f x)+1) \tanh ^{-1}\left (e^{i (e+f x)}\right )\right )}{\sqrt{2} a f \left (1+e^{i (e+f x)}\right ) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.3, size = 123, normalized size = 1.3 \begin{align*}{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}{4\,f{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 2\,\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +2\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +\cos \left ( fx+e \right ) -1 \right ){\frac{1}{\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 = 1.86656, size = 536, normalized size = 5.64 \begin{align*} -\frac{{\left ({\left (2 \,{\left (2 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (f x + e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, \sin \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (f x + e\right ) + 1\right )} \cos \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (f x + e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, \sin \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) - 1\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 ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt{a} \sqrt{c}}{2 \,{\left (a^{2} c \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a^{2} c \cos \left (f x + e\right )^{2} + a^{2} c \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a^{2} c \sin \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) + 4 \, a^{2} c \sin \left (f x + e\right )^{2} + 4 \, a^{2} c \cos \left (f x + e\right ) + a^{2} c + 2 \,{\left (2 \, a^{2} c \cos \left (f x + e\right ) + a^{2} c\right )} \cos \left (2 \, f x + 2 \, e\right )\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.665104, size = 952, normalized size = 10.02 \begin{align*} \left [-\frac{\sqrt{-a c}{\left (\cos \left (f x + e\right ) + 1\right )} \log \left (-\frac{4 \,{\left (2 \, \sqrt{-a c} \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 )^{2} +{\left (a c \cos \left (f x + e\right )^{2} + a c\right )} \sin \left (f x + e\right )\right )}}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, \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 )}{4 \,{\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c f\right )} \sin \left (f x + e\right )}, \frac{\sqrt{a c}{\left (\cos \left (f x + e\right ) + 1\right )} \arctan \left (\frac{\sqrt{a c} \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 )}}}{a c \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + \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 )}{2 \,{\left (a^{2} c f \cos \left (f x + e\right ) + a^{2} c f\right )} \sin \left (f x + e\right )}\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{\sec{\left (e + f x \right )}}{\left (a \left (\sec{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}} \sqrt{- c \left (\sec{\left (e + f x \right )} - 1\right )}}\, 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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