Integrand size = 31, antiderivative size = 167 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=-\frac {3 A \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (8+3 m),\frac {1}{6} (14+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (8+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {20, 2827, 2722} \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=-\frac {3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+5),\frac {1}{6} (3 m+11),\cos ^2(c+d x)\right )}{d (3 m+5) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+8),\frac {1}{6} (3 m+14),\cos ^2(c+d x)\right )}{d (3 m+8) \sqrt {\sin ^2(c+d x)}} \]
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Rule 20
Rule 2722
Rule 2827
Rubi steps \begin{align*} \text {integral}& = \frac {(b \cos (c+d x))^{2/3} \int \cos ^{\frac {2}{3}+m}(c+d x) (A+B \cos (c+d x)) \, dx}{\cos ^{\frac {2}{3}}(c+d x)} \\ & = \frac {\left (A (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{\cos ^{\frac {2}{3}}(c+d x)}+\frac {\left (B (b \cos (c+d x))^{2/3}\right ) \int \cos ^{\frac {5}{3}+m}(c+d x) \, dx}{\cos ^{\frac {2}{3}}(c+d x)} \\ & = -\frac {3 A \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (5+3 m) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (8+3 m),\frac {1}{6} (14+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (8+3 m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.84 \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=-\frac {3 \cos ^{1+m}(c+d x) (b \cos (c+d x))^{2/3} \csc (c+d x) \left (A (8+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5+3 m),\frac {1}{6} (11+3 m),\cos ^2(c+d x)\right )+B (5+3 m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (8+3 m),\frac {7}{3}+\frac {m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (5+3 m) (8+3 m)} \]
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\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}} \left (A +B \cos \left (d x +c \right )\right )d x\]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=\int \left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}} \left (A + B \cos {\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \]
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