Integrand size = 38, antiderivative size = 132 \[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=-\frac {B (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d n \sqrt {\sin ^2(c+d x)}}-\frac {C (b \cos (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 3089, 2827, 2722} \[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=-\frac {B \sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {n+2}{2},\cos ^2(c+d x)\right )}{d n \sqrt {\sin ^2(c+d x)}}-\frac {C \sin (c+d x) (b \cos (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(c+d x)\right )}{b d (n+1) \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rule 2827
Rule 3089
Rubi steps \begin{align*} \text {integral}& = b^2 \int (b \cos (c+d x))^{-2+n} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \\ & = b \int (b \cos (c+d x))^{-1+n} (B+C \cos (c+d x)) \, dx \\ & = (b B) \int (b \cos (c+d x))^{-1+n} \, dx+C \int (b \cos (c+d x))^n \, dx \\ & = -\frac {B (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 {C (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 \[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=-\frac {b (b \cos (c+d x))^{-1+n} \cot (c+d x) \left (B (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(c+d x)\right )+C n \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d n (1+n)} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )d x\]
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\[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Timed out} \]
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\[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{2} \,d x } \]
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\[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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