Integrand size = 25, antiderivative size = 70 \[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\frac {2 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)}} \]
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Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3891, 69, 67} \[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\frac {2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right )}{f \sqrt {a \sec (e+f x)+a}} \]
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Rule 67
Rule 69
Rule 3891
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \tan (e+f x)\right ) \text {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 {\left (a^2 (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \sin (e+f x)\right ) \text {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 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \sin (e+f x)}{f \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) (-\sec (e+f x))^n \sec ^{-n}(e+f x) \sqrt {a (1+\sec (e+f x))} \tan \left (\frac {1}{2} (e+f x)\right )}{f} \]
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\[\int \left (-\sec \left (f x +e \right )\right )^{n} \sqrt {a +a \sec \left (f x +e \right )}d x\]
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\[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\int \left (- \sec {\left (e + f x \right )}\right )^{n} \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}\, dx \]
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\[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} \left (-\sec \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (-\sec (e+f x))^n \sqrt {a+a \sec (e+f x)} \, dx=\int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (-\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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