Optimal. Leaf size=74 \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^m \text{Hypergeometric2F1}\left (3,m+\frac{1}{2},m+\frac{3}{2},\frac{1}{2} (\sec (e+f x)+1)\right )}{4 c^2 f (2 m+1) \sqrt{c-c \sec (e+f x)}} \]
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Rubi [A] time = 0.137036, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3961, 68} \[ -\frac{\tan (e+f x) (a \sec (e+f x)+a)^m \, _2F_1\left (3,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1)\right )}{4 c^2 f (2 m+1) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3961
Rule 68
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(a+a x)^{-\frac{1}{2}+m}}{(c-c x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{\, _2F_1\left (3,\frac{1}{2}+m;\frac{3}{2}+m;\frac{1}{2} (1+\sec (e+f x))\right ) (a+a \sec (e+f x))^m \tan (e+f x)}{4 c^2 f (1+2 m) \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [F] time = 2.03071, size = 0, normalized size = 0. \[ \int \frac{\sec (e+f x) (a+a \sec (e+f x))^m}{(c-c \sec (e+f x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int{\sec \left ( fx+e \right ) \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sec \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c \sec \left (f x + e\right ) + c}{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{c^{3} \sec \left (f x + e\right )^{3} - 3 \, c^{3} \sec \left (f x + e\right )^{2} + 3 \, c^{3} \sec \left (f x + e\right ) - c^{3}}, x\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sec \left (f x + e\right ) + a\right )}^{m} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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