Integrand size = 29, antiderivative size = 109 \[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(C (1+n)+A (2+n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n (2+n) \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^{1+n} \tan (c+d x)}{b d (2+n)} \]
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Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {16, 4131, 3857, 2722} \[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(A (n+2)+C (n+1)) \sin (c+d x) (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(c+d x)\right )}{d n (n+2) \sqrt {\sin ^2(c+d x)}}+\frac {C \tan (c+d x) (b \sec (c+d x))^{n+1}}{b d (n+2)} \]
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Rule 16
Rule 2722
Rule 3857
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \sec (c+d x))^{1+n} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b} \\ & = \frac {C (b \sec (c+d x))^{1+n} \tan (c+d x)}{b d (2+n)}+\frac {\left (A+\frac {C (1+n)}{2+n}\right ) \int (b \sec (c+d x))^{1+n} \, dx}{b} \\ & = \frac {C (b \sec (c+d x))^{1+n} \tan (c+d x)}{b d (2+n)}+\frac {\left (\left (A+\frac {C (1+n)}{2+n}\right ) \left (\frac {\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{-1-n} \, dx}{b} \\ & = \frac {\left (A+\frac {C (1+n)}{2+n}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {n}{2},\frac {2-n}{2},\cos ^2(c+d x)\right ) (b \sec (c+d x))^n \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^{1+n} \tan (c+d x)}{b d (2+n)} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\csc (c+d x) (b \sec (c+d x))^n \left (A (3+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\sec ^2(c+d x)\right )+C (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\sec ^2(c+d x)\right ) \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{d (1+n) (3+n)} \]
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\[\int \sec \left (d x +c \right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \sec \left (d x +c \right )^{2}\right )d x\]
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\[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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\[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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\[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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\[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \sec (c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{\cos \left (c+d\,x\right )} \,d x \]
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