Integrand size = 12, antiderivative size = 252 \[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=-\frac {3 \sqrt {3-i \sqrt {3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt {3-i \sqrt {3}}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right ) \sec (a+b x) \sqrt {1-\frac {(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt {\frac {i+\sqrt {3}}{3 i+\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt {3}\right ) c^{2/3}}} \sqrt {\frac {i-\sqrt {3}}{3 i-\sqrt {3}}+\frac {2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt {3}\right ) c^{2/3}}}}{\sqrt {2} b \sqrt [3]{c}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.23, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\frac {3 \cos (a+b x) (c \sin (a+b x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(a+b x)\right )}{2 b c \sqrt {\cos ^2(a+b x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{2/3}}{2 b c \sqrt {\cos ^2(a+b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\frac {3 \sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sin ^2(a+b x)\right ) \tan (a+b x)}{2 b \sqrt [3]{c \sin (a+b x)}} \]
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\[\int \frac {1}{\left (c \sin \left (b x +a \right )\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\int \frac {1}{\sqrt [3]{c \sin {\left (a + b x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{c \sin (a+b x)}} \, dx=\int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{1/3}} \,d x \]
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