Integrand size = 23, antiderivative size = 80 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 80, 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} \[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=-\frac {\cos (c+d x) \operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rule 129
Rule 440
Rule 2864
Rule 2866
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {\csc ^2(c+d x)}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}} \\ & = -\frac {\cos (c+d x) \text {Subst}\left (\int \frac {1}{(1-x)^2 (2-x)^{11/6} \sqrt {x}} \, dx,x,1-\sin (c+d x)\right )}{a d \sqrt {1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \\ & = -\frac {(2 \cos (c+d x)) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (2-x^2\right )^{11/6}} \, dx,x,\sqrt {1-\sin (c+d x)}\right )}{a d \sqrt {1-\sin (c+d x)} \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \\ & = -\frac {\operatorname {AppellF1}\left (\frac {1}{2},2,\frac {11}{6},\frac {3}{2},1-\sin (c+d x),\frac {1}{2} (1-\sin (c+d x))\right ) \cos (c+d x)}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}} \\ \end{align*}
\[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx \]
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\[\int \frac {\csc ^{2}\left (d x +c \right )}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}}d x\]
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Timed out. \[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {\csc ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {\csc \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int { \frac {\csc \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^2(c+d x)}{(a+a \sin (c+d x))^{4/3}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
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