Integrand size = 21, antiderivative size = 82 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 82, 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 = {20, 2722} \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+1),\frac {1}{6} (3 m+7),\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]
[In]
[Out]
Rule 20
Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}} \\ & = -\frac {3 \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) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{3}+m\right ),\frac {1}{2} \left (\frac {7}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {1}{3}+m\right ) (b \cos (c+d x))^{2/3}} \]
[In]
[Out]
\[\int \frac {\cos ^{m}\left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {2}{3}}}d x\]
[In]
[Out]
\[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {\cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
[In]
[Out]
\[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int { \frac {\cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^m}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
[In]
[Out]