Integrand size = 14, antiderivative size = 106 \[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=-\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{2/3}}{d \sqrt {1+\sin (c+d x)} (a+b \sin (c+d x))^{2/3}} \]
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Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2744, 144, 143} \[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=-\frac {\sqrt {2} \cos (c+d x) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{d \sqrt {\sin (c+d x)+1} (a+b \sin (c+d x))^{2/3}} \]
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Rule 143
Rule 144
Rule 2744
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} (a+b x)^{2/3}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}} \\ & = \frac {\left (\cos (c+d x) \left (-\frac {a+b \sin (c+d x)}{-a-b}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)} (a+b \sin (c+d x))^{2/3}} \\ & = -\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{2/3}}{d \sqrt {1+\sin (c+d x)} (a+b \sin (c+d x))^{2/3}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}} \sqrt [3]{a+b \sin (c+d x)}}{b d} \]
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\[\int \frac {1}{\left (a +b \sin \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\int { \frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b \sin (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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