Integrand size = 12, antiderivative size = 56 \[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\frac {3 c \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\sin ^2(a+b x)\right )}{5 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}} \]
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Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3857, 2722} \[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\frac {3 c \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\sin ^2(a+b x)\right )}{5 b \sqrt {\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}} \]
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Rule 2722
Rule 3857
Rubi steps \begin{align*} \text {integral}& = \sqrt [3]{c \csc (a+b x)} \sqrt [3]{\frac {\sin (a+b x)}{c}} \int \left (\frac {\sin (a+b x)}{c}\right )^{2/3} \, dx \\ & = \frac {3 \cos (a+b x) \sqrt [3]{c \csc (a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\sin ^2(a+b x)\right ) \sin ^2(a+b x)}{5 b c \sqrt {\cos ^2(a+b x)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=-\frac {\cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},\cos ^2(a+b x)\right ) \sin (a+b x)}{b (c \csc (a+b x))^{2/3} \sin ^2(a+b x)^{5/6}} \]
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\[\int \frac {1}{\left (c \csc \left (x b +a \right )\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c \csc (a+b x))^{2/3}} \, dx=\int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{2/3}} \,d x \]
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