Optimal. Leaf size=70 \[ \frac{2 a \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\sec (e+f x)\right )}{f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.0762857, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3806, 67, 65} \[ \frac{2 a \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left (\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right )}{f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3806
Rule 67
Rule 65
Rubi steps
\begin{align*} \int (d \sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx &=-\frac{\left (a^2 d \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(d 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{\left (a^2 \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \sin (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 ) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \sin (e+f x)}{f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.119744, size = 71, normalized size = 1.01 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \sec ^{-n}(e+f x) (d \sec (e+f x))^n \text{Hypergeometric2F1}\left (\frac{1}{2},1-n,\frac{3}{2},1-\sec (e+f x)\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int \left ( d\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 (d \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 (d \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 \sqrt{a \left (\sec{\left (e + f x \right )} + 1\right )} \left (d \sec{\left (e + f x \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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