Integrand size = 10, antiderivative size = 53 \[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {3 \cos ^{\frac {2}{3}}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(a+b x)\right ) \sin (a+b x)}{2 b \sqrt {\sin ^2(a+b x)}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {3 \sin (a+b x) \cos ^{\frac {2}{3}}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(a+b x)\right )}{2 b \sqrt {\sin ^2(a+b x)}} \]
[In]
[Out]
Rule 2722
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \cos ^{\frac {2}{3}}(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(a+b x)\right ) \sin (a+b x)}{2 b \sqrt {\sin ^2(a+b x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {3 \cos ^{\frac {2}{3}}(a+b x) \csc (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{2 b} \]
[In]
[Out]
\[\int \frac {1}{\cos \left (b x +a \right )^{\frac {1}{3}}}d x\]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=\int \frac {1}{\sqrt [3]{\cos {\left (a + b x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=\int { \frac {1}{\cos \left (b x + a\right )^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
Time = 14.95 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt [3]{\cos (a+b x)}} \, dx=-\frac {3\,{\cos \left (a+b\,x\right )}^{2/3}\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {4}{3};\ {\cos \left (a+b\,x\right )}^2\right )}{2\,b\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]
[In]
[Out]