Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2731, 2730} \[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rule 2730
Rule 2731
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {1}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}} \\ & = -\frac {\cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sqrt {2-2 \sin (c+d x)}-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right ) (1+\sin (c+d x))\right )}{5 d \sqrt {2-2 \sin (c+d x)} (a (1+\sin (c+d x)))^{4/3}} \]
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\[\int \frac {1}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {1}{\left (a \sin {\left (c + d x \right )} + a\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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