Integrand size = 40, antiderivative size = 163 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 B \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+2 n) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+2 n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {20, 3089, 2827, 2722} \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n+5),\frac {1}{4} (2 n+9),\cos ^2(c+d x)\right )}{d (2 n+5) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n+7),\frac {1}{4} (2 n+11),\cos ^2(c+d x)\right )}{d (2 n+7) \sqrt {\sin ^2(c+d x)}} \]
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Rule 20
Rule 2722
Rule 2827
Rule 3089
Rubi steps \begin{align*} \text {integral}& = \left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {1}{2}+n}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \\ & = \left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {3}{2}+n}(c+d x) (B+C \cos (c+d x)) \, dx \\ & = \left (B \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {3}{2}+n}(c+d x) \, dx+\left (C \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{\frac {5}{2}+n}(c+d x) \, dx \\ & = -\frac {2 B \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+2 n) \sqrt {\sin ^2(c+d x)}}-\frac {2 C \cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+2 n) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^n \csc (c+d x) \left (B (7+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (5+2 n),\frac {1}{4} (9+2 n),\cos ^2(c+d x)\right )+C (5+2 n) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (5+2 n) (7+2 n)} \]
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\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )d x\]
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\[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sqrt {\cos \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right ) \,d x \]
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