Integrand size = 25, antiderivative size = 95 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3093, 2722} \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b d}-\frac {3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right )}{10 b d \sqrt {\sin ^2(c+d x)}} \]
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Rule 2722
Rule 3093
Rubi steps \begin{align*} \text {integral}& = \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b d}+\frac {1}{5} (5 A+2 C) \int \frac {1}{\sqrt [3]{b \cos (c+d x)}} \, dx \\ & = \frac {3 C (b \cos (c+d x))^{2/3} \sin (c+d x)}{5 b d}-\frac {3 (5 A+2 C) (b \cos (c+d x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 b d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=-\frac {3 \cot (c+d x) \left (4 A \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right )+C \cos ^2(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{8 d \sqrt [3]{b \cos (c+d x)}} \]
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\[\int \frac {A +C \left (\cos ^{2}\left (d x +c \right )\right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \]
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