Integrand size = 10, antiderivative size = 51 \[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=-\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3857, 2722} \[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=-\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \]
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Rule 2722
Rule 3857
Rubi steps \begin{align*} \text {integral}& = \sqrt [3]{\csc (a+b x)} \sqrt [3]{\sin (a+b x)} \int \frac {1}{\sin ^{\frac {4}{3}}(a+b x)} \, dx \\ & = -\frac {3 \cos (a+b x) \sqrt [3]{\csc (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b \sqrt {\cos ^2(a+b x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\frac {\cos (a+b x) \sqrt [3]{\csc (a+b x)} \left (-3+2 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {3}{2},\cos ^2(a+b x)\right ) \sqrt [6]{\sin ^2(a+b x)}\right )}{b} \]
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\[\int \csc \left (x b +a \right )^{\frac {4}{3}}d x\]
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\[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {4}{3}} \,d x } \]
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\[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\int \csc ^{\frac {4}{3}}{\left (a + b x \right )}\, dx \]
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\[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {4}{3}} \,d x } \]
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\[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {4}{3}} \,d x } \]
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Timed out. \[ \int \csc ^{\frac {4}{3}}(a+b x) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{4/3} \,d x \]
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