Optimal. Leaf size=104 \[ -\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^2 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}+\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^2 f} \]
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Rubi [A] time = 0.157338, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3904, 3887, 325, 203} \[ -\frac{2 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{3 a c^2 f}+\frac{2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c^2 f}+\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c^2 f} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sec (e+f x)}}{(c-c \sec (e+f x))^2} \, dx &=\frac{\int \cot ^4(e+f x) (a+a \sec (e+f x))^{5/2} \, dx}{a^2 c^2}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{a c^2 f}\\ &=-\frac{2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^2 f}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^2 f}\\ &=\frac{2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^2 f}-\frac{2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^2 f}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^2 f}\\ &=\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c^2 f}+\frac{2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c^2 f}-\frac{2 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{3 a c^2 f}\\ \end{align*}
Mathematica [C] time = 0.241069, size = 78, normalized size = 0.75 \[ -\frac{2 \sqrt{\cos (e+f x)} \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )}{3 c^2 f (\cos (e+f x)-1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.289, size = 214, normalized size = 2.1 \begin{align*} -{\frac{1}{3\,f{c}^{2}\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) }\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 3\,\sqrt{2}\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) -3\,\sqrt{2}\sin \left ( fx+e \right ){\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) \sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}-8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+6\,\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sec \left (f x + e\right ) + a}}{{\left (c \sec \left (f x + e\right ) - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44907, size = 872, normalized size = 8.38 \begin{align*} \left [\frac{3 \, \sqrt{-a}{\left (\cos \left (f x + e\right ) - 1\right )} \log \left (-\frac{8 \, a \cos \left (f x + e\right )^{3} - 4 \,{\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 4 \,{\left (4 \, \cos \left (f x + e\right )^{2} - 3 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{6 \,{\left (c^{2} f \cos \left (f x + e\right ) - c^{2} f\right )} \sin \left (f x + e\right )}, \frac{3 \, \sqrt{a}{\left (\cos \left (f x + e\right ) - 1\right )} \arctan \left (\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \,{\left (4 \, \cos \left (f x + e\right )^{2} - 3 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{3 \,{\left (c^{2} f \cos \left (f x + e\right ) - c^{2} 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*} \frac{\int \frac{\sqrt{a \sec{\left (e + f x \right )} + a}}{\sec ^{2}{\left (e + f x \right )} - 2 \sec{\left (e + f x \right )} + 1}\, dx}{c^{2}} \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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