Integrand size = 41, antiderivative size = 154 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^4 d}-\frac {3 (11 A+8 C) (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{88 b^4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{11/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{11 b^5 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.16 (sec) , antiderivative size = 154, 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 = {16, 3102, 2827, 2722} \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=-\frac {3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right )}{88 b^4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{11/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\cos ^2(c+d x)\right )}{11 b^5 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^4 d} \]
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Rule 16
Rule 2722
Rule 2827
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{5/3} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b^3} \\ & = \frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^4 d}+\frac {3 \int (b \cos (c+d x))^{5/3} \left (\frac {1}{3} b (11 A+8 C)+\frac {11}{3} b B \cos (c+d x)\right ) \, dx}{11 b^4} \\ & = \frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^4 d}+\frac {B \int (b \cos (c+d x))^{8/3} \, dx}{b^4}+\frac {(11 A+8 C) \int (b \cos (c+d x))^{5/3} \, dx}{11 b^3} \\ & = \frac {3 C (b \cos (c+d x))^{8/3} \sin (c+d x)}{11 b^4 d}-\frac {3 (11 A+8 C) (b \cos (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{88 b^4 d \sqrt {\sin ^2(c+d x)}}-\frac {3 B (b \cos (c+d x))^{11/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{11 b^5 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=-\frac {3 \cos ^3(c+d x) \cot (c+d x) \left (-8 C \sin ^2(c+d x)+(11 A+8 C) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}+8 B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{88 d (b \cos (c+d x))^{4/3}} \]
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\[\int \frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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