Integrand size = 23, antiderivative size = 117 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}-\frac {(C (1+n)+A (2+n)) (b \cos (c+d x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (1+n) (2+n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3093, 2722} \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C \sin (c+d x) (b \cos (c+d x))^{n+1}}{b d (n+2)}-\frac {(A (n+2)+C (n+1)) \sin (c+d x) (b \cos (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(c+d x)\right )}{b d (n+1) (n+2) \sqrt {\sin ^2(c+d x)}} \]
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Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = \frac {C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}+\left (A+\frac {C (1+n)}{2+n}\right ) \int (b \cos (c+d x))^n \, dx \\ & = \frac {C (b \cos (c+d x))^{1+n} \sin (c+d x)}{b d (2+n)}-\frac {\left (A+\frac {C (1+n)}{2+n}\right ) (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 = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(b \cos (c+d x))^n \cot (c+d x) \left (A (3+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(c+d x)\right )+C (1+n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+n}{2},\frac {5+n}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+n) (3+n)} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
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\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )\, dx \]
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\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
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\[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]
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