Optimal. Leaf size=68 \[ \frac{2 a \tan (e+f x) (c-c \sec (e+f x))^n \text{Hypergeometric2F1}\left (1,n+\frac{1}{2},n+\frac{3}{2},1-\sec (e+f x)\right )}{f (2 n+1) \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.0827766, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3912, 65} \[ \frac{2 a \tan (e+f x) (c-c \sec (e+f x))^n \, _2F_1\left (1,n+\frac{1}{2};n+\frac{3}{2};1-\sec (e+f x)\right )}{f (2 n+1) \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3912
Rule 65
Rubi steps
\begin{align*} \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx &=-\frac{(a c \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(c-c x)^{-\frac{1}{2}+n}}{x} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=\frac{2 a \, _2F_1\left (1,\frac{1}{2}+n;\frac{3}{2}+n;1-\sec (e+f x)\right ) (c-c \sec (e+f x))^n \tan (e+f x)}{f (1+2 n) \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [F] time = 0.146506, size = 0, normalized size = 0. \[ \int \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.302, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+a\sec \left ( fx+e \right ) } \left ( c-c\sec \left ( fx+e \right ) \right ) ^{n}\, dx \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 \sqrt{a \sec \left (f x + e\right ) + a}{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}\,{d x} \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 (\sqrt{a \sec \left (f x + e\right ) + a}{\left (-c \sec \left (f x + e\right ) + c\right )}^{n}, x\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 \sqrt{a \left (\sec{\left (e + f x \right )} + 1\right )} \left (- c \left (\sec{\left (e + f x \right )} - 1\right )\right )^{n}\, 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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