Integrand size = 23, antiderivative size = 77 \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},2,-\frac {1}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{d (1+\sin (c+d x))^{7/6}} \]
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Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2866, 2864, 129, 440} \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=-\frac {2 \sqrt [6]{2} \cos (c+d x) (a \sin (c+d x)+a)^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},2,-\frac {1}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{7/6}} \]
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Rule 129
Rule 440
Rule 2864
Rule 2866
Rubi steps \begin{align*} \text {integral}& = \frac {(a+a \sin (c+d x))^{2/3} \int \csc ^2(c+d x) (1+\sin (c+d x))^{2/3} \, dx}{(1+\sin (c+d x))^{2/3}} \\ & = -\frac {\left (\cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{2-x}}{(1-x)^2 \sqrt {x}} \, dx,x,1-\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}} \\ & = -\frac {\left (2 \cos (c+d x) (a+a \sin (c+d x))^{2/3}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{2-x^2}}{\left (1-x^2\right )^2} \, dx,x,\sqrt {1-\sin (c+d x)}\right )}{d \sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{7/6}} \\ & = -\frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},2,-\frac {1}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x) (a+a \sin (c+d x))^{2/3}}{d (1+\sin (c+d x))^{7/6}} \\ \end{align*}
\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx \]
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\[\int \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
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Timed out. \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\text {Timed out} \]
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\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {2}{3}} \csc ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \csc \left (d x + c\right )^{2} \,d x } \]
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\[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \csc \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \csc ^2(c+d x) (a+a \sin (c+d x))^{2/3} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{2/3}}{{\sin \left (c+d\,x\right )}^2} \,d x \]
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