Optimal. Leaf size=47 \[ \frac{2 a \tan (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},\sec (e+f x)+1\right )}{f \sqrt{a-a \sec (e+f x)}} \]
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Rubi [A] time = 0.0718413, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3806, 65} \[ \frac{2 a \tan (e+f x) \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right )}{f \sqrt{a-a \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3806
Rule 65
Rubi steps
\begin{align*} \int (-\sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx &=\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(-x)^{-1+n}}{\sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1+\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt{a-a \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 70.3183, size = 213, normalized size = 4.53 \[ \frac{2^{n-\frac{1}{2}} e^{\frac{1}{2} i (e+f (1-2 n) x)} \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^{n-\frac{1}{2}} \csc \left (\frac{e}{2}+\frac{f x}{2}\right ) \sqrt{a-a \sec (e+f x)} (-\sec (e+f x))^n \sec ^{-n-\frac{1}{2}}(e+f x) \left ((n+1) e^{i f n x} \text{Hypergeometric2F1}\left (1,\frac{1-n}{2},\frac{n+2}{2},-e^{2 i (e+f x)}\right )-n e^{i (e+f (n+1) x)} \text{Hypergeometric2F1}\left (1,1-\frac{n}{2},\frac{n+3}{2},-e^{2 i (e+f x)}\right )\right )}{f n (n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.199, size = 0, normalized size = 0. \begin{align*} \int \left ( -\sec \left ( fx+e \right ) \right ) ^{n}\sqrt{a-a\sec \left ( fx+e \right ) }\, 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 (-\sec \left (f x + e\right )\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 (-\sec \left (f x + e\right )\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 \left (- \sec{\left (e + f x \right )}\right )^{n} \sqrt{- a \left (\sec{\left (e + f x \right )} - 1\right )}\, dx \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 \sqrt{-a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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