Integrand size = 21, antiderivative size = 45 \[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt {1+\sec (e+f x)}} \]
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Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3891, 67} \[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\frac {2 \tan (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right )}{f \sqrt {\sec (e+f x)+1}} \]
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Rule 67
Rule 3891
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (e+f x) \text {Subst}\left (\int \frac {x^{-1+n}}{\sqrt {1-x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}} \\ & = \frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt {1+\sec (e+f x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) \tan (e+f x)}{f \sqrt {1+\sec (e+f x)}} \]
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\[\int \sec \left (f x +e \right )^{n} \sqrt {\sec \left (f x +e \right )+1}d x\]
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\[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\int { \sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1} \,d x } \]
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\[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\int \sqrt {\sec {\left (e + f x \right )} + 1} \sec ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\int { \sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1} \,d x } \]
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\[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\int { \sec \left (f x + e\right )^{n} \sqrt {\sec \left (f x + e\right ) + 1} \,d x } \]
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Timed out. \[ \int \sec ^n(e+f x) \sqrt {1+\sec (e+f x)} \, dx=\int \sqrt {\frac {1}{\cos \left (e+f\,x\right )}+1}\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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