Optimal. Leaf size=58 \[ \frac{\tan (e+f x) F_1\left (\frac{1}{2};1-n,1;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right )}{f \sqrt{a-a \sec (e+f x)}} \]
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Rubi [A] time = 0.154118, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3828, 3825, 130, 429} \[ \frac{\tan (e+f x) F_1\left (\frac{1}{2};1-n,1;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right )}{f \sqrt{a-a \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3828
Rule 3825
Rule 130
Rule 429
Rubi steps
\begin{align*} \int \frac{(-\sec (e+f x))^n}{\sqrt{a-a \sec (e+f x)}} \, dx &=\frac{\sqrt{1-\sec (e+f x)} \int \frac{(-\sec (e+f x))^n}{\sqrt{1-\sec (e+f x)}} \, dx}{\sqrt{a-a \sec (e+f x)}}\\ &=\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{(1-x)^{-1+n}}{(2-x) \sqrt{x}} \, dx,x,1+\sec (e+f x)\right )}{f \sqrt{1+\sec (e+f x)} \sqrt{a-a \sec (e+f x)}}\\ &=\frac{(2 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^{-1+n}}{2-x^2} \, dx,x,\sqrt{1+\sec (e+f x)}\right )}{f \sqrt{1+\sec (e+f x)} \sqrt{a-a \sec (e+f x)}}\\ &=\frac{F_1\left (\frac{1}{2};1-n,1;\frac{3}{2};1+\sec (e+f x),\frac{1}{2} (1+\sec (e+f x))\right ) \tan (e+f x)}{f \sqrt{a-a \sec (e+f x)}}\\ \end{align*}
Mathematica [F] time = 1.24638, size = 0, normalized size = 0. \[ \int \frac{(-\sec (e+f x))^n}{\sqrt{a-a \sec (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.179, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -\sec \left ( fx+e \right ) \right ) ^{n}{\frac{1}{\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 \frac{\left (-\sec \left (f x + e\right )\right )^{n}}{\sqrt{-a \sec \left (f x + e\right ) + a}}\,{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 (-\frac{\sqrt{-a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n}}{a \sec \left (f x + e\right ) - a}, 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 \frac{\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 \frac{\left (-\sec \left (f x + e\right )\right )^{n}}{\sqrt{-a \sec \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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