Integrand size = 21, antiderivative size = 230 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}-\frac {a^3 (1+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (7+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n (2+n) \sqrt {\sin ^2(e+f x)}} \]
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Time = 0.35 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3899, 4082, 3872, 3857, 2722} \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=-\frac {a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(e+f x)\right )}{f \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(e+f x)\right )}{f n (n+2) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac {\sin (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) \sec ^{n+1}(e+f x)}{f (n+2)} \]
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Rule 2722
Rule 3857
Rule 3872
Rule 3899
Rule 4082
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {a \int \sec ^n(e+f x) (a+a \sec (e+f x)) (a (2+2 n)+a (5+2 n) \sec (e+f x)) \, dx}{2+n} \\ & = \frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {a \int \sec ^n(e+f x) \left (a^2 (2+n) (1+4 n)+a^2 (1+n) (7+4 n) \sec (e+f x)\right ) \, dx}{2+3 n+n^2} \\ & = \frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {\left (a^3 (1+4 n)\right ) \int \sec ^n(e+f x) \, dx}{1+n}+\frac {\left (a^3 (7+4 n)\right ) \int \sec ^{1+n}(e+f x) \, dx}{2+n} \\ & = \frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {\left (a^3 (1+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx}{1+n}+\frac {\left (a^3 (7+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-1-n}(e+f x) \, dx}{2+n} \\ & = \frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}-\frac {a^3 (1+4 n) \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 \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (7+4 n) \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 (2+n) \sqrt {\sin ^2(e+f x)}} \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.72 \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\frac {a^3 \csc (e+f x) \sec ^{-1+n}(e+f x) \left (n (3 (2+n)+(1+n) \sec (e+f x)) \tan ^2(e+f x)+\left (2+9 n+4 n^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\sec ^2(e+f x)\right ) \sqrt {-\tan ^2(e+f x)}+n (7+4 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sec ^2(e+f x)\right ) \sec (e+f x) \sqrt {-\tan ^2(e+f x)}\right )}{f n (1+n) (2+n)} \]
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\[\int \sec \left (f x +e \right )^{n} \left (a +a \sec \left (f x +e \right )\right )^{3}d x\]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=a^{3} \left (\int 3 \sec {\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int 3 \sec ^{2}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{3}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{n}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sec \left (f x + e\right )^{n} \,d x } \]
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\[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{3} \sec \left (f x + e\right )^{n} \,d x } \]
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Timed out. \[ \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]
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