Integrand size = 33, antiderivative size = 136 \[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt {\cos (c+d x)}}+\frac {2 (A-C (1-2 n)+2 A n) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+2 n),\frac {1}{4} (3+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d \left (1-4 n^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {20, 3093, 2722} \[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2 n-1),\frac {1}{4} (2 n+3),\cos ^2(c+d x)\right )}{d \left (1-4 n^2\right ) \sqrt {\sin ^2(c+d x)} \sqrt {\cos (c+d x)}}+\frac {2 C \sin (c+d x) (b \cos (c+d x))^n}{d (2 n+1) \sqrt {\cos (c+d x)}} \]
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Rule 20
Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = \left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {3}{2}+n}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt {\cos (c+d x)}}+\frac {\left (\left (C \left (-\frac {1}{2}+n\right )+A \left (\frac {1}{2}+n\right )\right ) \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {3}{2}+n}(c+d x) \, dx}{\frac {1}{2}+n} \\ & = \frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (1+2 n) \sqrt {\cos (c+d x)}}+\frac {2 (A-C (1-2 n)+2 A n) (b \cos (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+2 n),\frac {1}{4} (3+2 n),\cos ^2(c+d x)\right ) \sin (c+d x)}{d \left (1-4 n^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.03 \[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 (b \cos (c+d x))^n \csc (c+d x) \left (A (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1+2 n),\frac {1}{4} (3+2 n),\cos ^2(c+d x)\right )+C (-1+2 n) \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3+2 n),\frac {1}{4} (7+2 n),\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (-1+2 n) (3+2 n) \sqrt {\cos (c+d x)}} \]
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\[\int \frac {\left (\cos \left (d x +c \right ) b \right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\cos \left (d x +c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (b \cos {\left (c + d x \right )}\right )^{n} \left (A + C \cos ^{2}{\left (c + d x \right )}\right )}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{n}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^{3/2}} \,d x \]
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