Optimal. Leaf size=181 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{4 \sqrt{2} a^3 \sqrt{c} f}+\frac{\tan (e+f x)}{4 f \left (a^3 \sec (e+f x)+a^3\right ) \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.394111, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {3960, 3795, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{4 \sqrt{2} a^3 \sqrt{c} f}+\frac{\tan (e+f x)}{4 f \left (a^3 \sec (e+f x)+a^3\right ) \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{5 f (a \sec (e+f x)+a)^3 \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3960
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}} \, dx &=\frac{\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}} \, dx}{2 a}\\ &=\frac{\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) \sqrt{c-c \sec (e+f x)}} \, dx}{4 a^2}\\ &=\frac{\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}+\frac{\int \frac{\sec (e+f x)}{\sqrt{c-c \sec (e+f x)}} \, dx}{8 a^3}\\ &=\frac{\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{2 c+x^2} \, dx,x,\frac{c \tan (e+f x)}{\sqrt{c-c \sec (e+f x)}}\right )}{4 a^3 f}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{4 \sqrt{2} a^3 \sqrt{c} f}+\frac{\tan (e+f x)}{5 f (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{6 a f (a+a \sec (e+f x))^2 \sqrt{c-c \sec (e+f x)}}+\frac{\tan (e+f x)}{4 f \left (a^3+a^3 \sec (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.58037, size = 225, normalized size = 1.24 \[ \frac{2 e^{-\frac{1}{2} i (e+f x)} \sin \left (\frac{1}{2} (e+f x)\right ) \cos \left (\frac{1}{2} (e+f x)\right ) \sec ^{\frac{7}{2}}(e+f x) \left (\frac{e^{\frac{1}{2} i (e+f x)} (80 \cos (e+f x)+37 \cos (2 (e+f x))+67)}{8 \sqrt{\sec (e+f x)}}-15 \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \cos ^5\left (\frac{1}{2} (e+f x)\right ) \tanh ^{-1}\left (\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right )\right )}{15 a^3 f (\sec (e+f x)+1)^3 \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.244, size = 155, normalized size = 0.9 \begin{align*} -{\frac{-1+\cos \left ( fx+e \right ) }{60\,f{a}^{3}\sin \left ( fx+e \right ) } \left ( 3\, \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{5/2}-5\, \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{3/2}+15\,\arctan \left ({\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}}} \right ) +15\,\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}} \right ){\frac{1}{\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}}}{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \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{\sec \left (f x + e\right )}{{\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sqrt{-c \sec \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.659831, size = 1053, normalized size = 5.82 \begin{align*} \left [-\frac{15 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt{-c} \log \left (\frac{2 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{-c} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} +{\left (3 \, c \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 4 \,{\left (37 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 15 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{240 \,{\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c f\right )} \sin \left (f x + e\right )}, \frac{15 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt{c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \,{\left (37 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} + 15 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{120 \,{\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} 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*} \frac{\int \frac{\sec{\left (e + f x \right )}}{\sqrt{- c \sec{\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )} + 3 \sqrt{- c \sec{\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )} + 3 \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )} + \sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.61584, size = 261, normalized size = 1.44 \begin{align*} -\frac{\frac{\sqrt{2}{\left (15 i \, \sqrt{-c} \arctan \left (-i\right ) - 23 \, \sqrt{-c}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}{a^{3} c} + \frac{\sqrt{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{\sqrt{c}}\right )}{\sqrt{c}} - \frac{3 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{5}{2}} c^{12} - 5 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} c^{13} + 15 \, \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c^{14}}{c^{15}}\right )}}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{120 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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