Integrand size = 27, antiderivative size = 132 \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=-\frac {A (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {16, 2827, 2722} \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=-\frac {A \sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {n+2}{2},\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}}-\frac {B \sin (c+d x) (b \cos (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(c+d x)\right )}{b d (n+1) \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rule 2827
Rubi steps \begin{align*} \text {integral}& = b \int (b \cos (c+d x))^{-1+n} (A+B \cos (c+d x)) \, dx \\ & = (A b) \int (b \cos (c+d x))^{-1+n} \, dx+B \int (b \cos (c+d x))^n \, dx \\ & = -\frac {A (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}-\frac {B (b \cos (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=-\frac {b (b \cos (c+d x))^{-1+n} \cot (c+d x) \left (A (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right )+B n \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d n (1+n)} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +B \cos \left (d x +c \right )\right ) \sec \left (d x +c \right )d x\]
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\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + B \cos {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{\cos \left (c+d\,x\right )} \,d x \]
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