Optimal. Leaf size=103 \[ \frac{8 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.171966, antiderivative size = 103, 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, 461, 203} \[ \frac{8 a^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a}}{c f}+\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 461
Rule 203
Rubi steps
\begin{align*} \int \frac{(a+a \sec (e+f x))^{5/2}}{c-c \sec (e+f x)} \, dx &=-\frac{\int \cot ^2(e+f x) (a+a \sec (e+f x))^{7/2} \, dx}{a c}\\ &=\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{\left (2+a x^2\right )^2}{x^2 \left (1+a x^2\right )} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c f}\\ &=\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (a+\frac{4}{x^2}-\frac{a}{1+a x^2}\right ) \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c f}\\ &=\frac{8 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a+a \sec (e+f x)}}-\frac{\left (2 a^3\right ) \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 f}\\ &=\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{c f}+\frac{8 a^2 \cot (e+f x) \sqrt{a+a \sec (e+f x)}}{c f}-\frac{2 a^3 \tan (e+f x)}{c f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.680392, size = 96, normalized size = 0.93 \[ \frac{2 a^3 \tan (e+f x) \sec (e+f x) \left ((5 \cos (e+f x)-1) \sqrt{\sec (e+f x)-1}-(\cos (e+f x)-1) \tan ^{-1}\left (\sqrt{\sec (e+f x)-1}\right )\right )}{c f (\sec (e+f x)-1)^{3/2} \sqrt{a (\sec (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.227, size = 120, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2}}{fc\sin \left ( fx+e \right ) } \left ( \sqrt{2}\sin \left ( fx+e \right ){\it Artanh} \left ({\frac{\sqrt{2}\sin \left ( fx+e \right ) }{2\,\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 ) }}}-10\,\cos \left ( fx+e \right ) +2 \right ) \sqrt{{\frac{a \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}{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 = 1.47104, size = 724, normalized size = 7.03 \begin{align*} \left [\frac{\sqrt{-a} a^{2} \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 (5 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{2 \, c f \sin \left (f x + e\right )}, \frac{a^{\frac{5}{2}} \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 (5 \, a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{c f \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(-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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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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