Optimal. Leaf size=46 \[ \frac{\tan (e+f x) \log (\tan (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.255427, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {3963, 2620, 29} \[ \frac{\tan (e+f x) \log (\tan (e+f x))}{f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3963
Rule 2620
Rule 29
Rubi steps
\begin{align*} \int \frac{\sec ^2(e+f x)}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}} \, dx &=\frac{\tan (e+f x) \int \csc (e+f x) \sec (e+f x) \, dx}{\sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tan (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{\log (\tan (e+f x)) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.788904, size = 94, normalized size = 2.04 \[ \frac{4 i \left (-1+e^{i (e+f x)}\right ) \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \tanh ^{-1}\left (e^{2 i (e+f x)}\right )}{f \left (1+e^{i (e+f x)}\right ) \sqrt{a (\sec (e+f x)+1)} \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.276, size = 138, normalized size = 3. \begin{align*}{\frac{\cos \left ( fx+e \right ) }{af\sin \left ( fx+e \right ) c}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( \ln \left ({\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \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 [A] time = 1.88718, size = 76, normalized size = 1.65 \begin{align*} -\frac{\arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) - 1\right )}{\sqrt{a} \sqrt{c} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.735096, size = 613, normalized size = 13.33 \begin{align*} \left [-\frac{\sqrt{-a c} \log \left (-\frac{8 \,{\left ({\left (2 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )\right )} \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 )}} +{\left (2 \, a c \cos \left (f x + e\right )^{4} - 2 \, 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 )^{4} - \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right )}{2 \, a c f}, \frac{\sqrt{a c} \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 )}} \cos \left (f x + e\right )}{{\left (2 \, a c \cos \left (f x + e\right )^{2} - a c\right )} \sin \left (f x + e\right )}\right )}{a c f}\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 ^{2}{\left (e + f x \right )}}{\sqrt{a \left (\sec{\left (e + f x \right )} + 1\right )} \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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