Integrand size = 21, antiderivative size = 58 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {16, 2722} \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \sin (c+d x) (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right )}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \]
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Rule 16
Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{4/3} \, dx}{b^2} \\ & = -\frac {3 (b \cos (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b^3 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^2(c+d x) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{7 d (b \cos (c+d x))^{2/3}} \]
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\[\int \frac {\cos ^{2}\left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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