Integrand size = 23, antiderivative size = 108 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\frac {2 a^2 \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (1+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}} \]
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Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3899, 21, 3891, 67} \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\frac {2 a^2 (4 n+1) \tan (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right )}{f (2 n+1) \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt {a \sec (e+f x)+a}} \]
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Rule 21
Rule 67
Rule 3891
Rule 3899
Rubi steps \begin{align*} \text {integral}& = \frac {2 a^2 \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {(2 a) \int \frac {\sec ^n(e+f x) \left (a \left (\frac {1}{2}+2 n\right )+a \left (\frac {1}{2}+2 n\right ) \sec (e+f x)\right )}{\sqrt {a+a \sec (e+f x)}} \, dx}{1+2 n} \\ & = \frac {2 a^2 \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {(a (1+4 n)) \int \sec ^n(e+f x) \sqrt {a+a \sec (e+f x)} \, dx}{1+2 n} \\ & = \frac {2 a^2 \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^3 (1+4 n) \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 (1+2 n) \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^2 \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (1+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-n,\frac {3}{2},1-\sec (e+f x)\right ) \tan (e+f x)}{f (1+2 n) \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\frac {a \left (-1+(1+4 n) \cos ^{\frac {1}{2}+n}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}+n,\frac {3}{2},2 \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^n(e+f x) \sqrt {a (1+\sec (e+f x))} \tan \left (\frac {1}{2} (e+f x)\right )}{f n} \]
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\[\int \sec \left (f x +e \right )^{n} \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sec ^{n}{\left (e + f x \right )}\, dx \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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