Integrand size = 41, antiderivative size = 227 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac {3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {20, 3102, 2827, 2722} \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+1),\frac {1}{6} (3 m+7),\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac {3 B \sin (c+d x) \cos ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+4),\frac {1}{6} (3 m+10),\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}+\frac {3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}} \]
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Rule 20
Rule 2722
Rule 2827
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{-\frac {2}{3}+m}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{(b \cos (c+d x))^{2/3}} \\ & = \frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac {\left (3 \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \left (\frac {1}{3} \left (3 C \left (\frac {1}{3}+m\right )+3 A \left (\frac {4}{3}+m\right )\right )+\frac {1}{3} B (4+3 m) \cos (c+d x)\right ) \, dx}{(4+3 m) (b \cos (c+d x))^{2/3}} \\ & = \frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}+\frac {\left (B \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{\frac {1}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}}+\frac {\left ((C+3 C m+A (4+3 m)) \cos ^{\frac {2}{3}}(c+d x)\right ) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{(4+3 m) (b \cos (c+d x))^{2/3}} \\ & = \frac {3 C \cos ^{1+m}(c+d x) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3}}-\frac {3 (C+3 C m+A (4+3 m)) \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {1}{6} (10+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\frac {3 \cos ^{1+m}(c+d x) \csc (c+d x) \left (-\left ((C+3 C m+A (4+3 m)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )+(1+3 m) \left (C \sin ^2(c+d x)-B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )\right )}{d (1+3 m) (4+3 m) (b \cos (c+d x))^{2/3}} \]
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\[\int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^m(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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