Integrand size = 19, antiderivative size = 132 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=-\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f (1-n) \sqrt {\sin ^2(e+f x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3872, 3857, 2722} \[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\frac {a \sin (e+f x) \sec ^n(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right )}{f n \sqrt {\sin ^2(e+f x)}}-\frac {a \sin (e+f x) \sec ^{n-1}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f (1-n) \sqrt {\sin ^2(e+f x)}} \]
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Rule 2722
Rule 3857
Rule 3872
Rubi steps \begin{align*} \text {integral}& = a \int \sec ^n(e+f x) \, dx+a \int \sec ^{1+n}(e+f x) \, dx \\ & = \left (a \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-1-n}(e+f x) \, dx+\left (a \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx \\ & = -\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f (1-n) \sqrt {\sin ^2(e+f x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.80 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\frac {a \csc (e+f x) \sec ^{-1+n}(e+f x) \left ((1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\sec ^2(e+f x)\right )+n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sec ^2(e+f x)\right ) \sec (e+f x)\right ) \sqrt {-\tan ^2(e+f x)}}{f n (1+n)} \]
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\[\int \sec \left (f x +e \right )^{n} \left (a +a \sec \left (f x +e \right )\right )d x\]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=a \left (\int \sec {\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{n}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \sec \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx=\int \left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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