Integrand size = 23, antiderivative size = 222 \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{27 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (3+3 \csc (e+f x))^2}+\frac {2 n \cos (e+f x) (d \csc (e+f x))^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-n),-\frac {n}{2},\sin ^2(e+f x)\right )}{27 d^2 f \sqrt {\cos ^2(e+f x)}}-\frac {(1+2 n) \cos (e+f x) (d \csc (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-n),\frac {1-n}{2},\sin ^2(e+f x)\right )}{27 d f \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3317, 3902, 4105, 3872, 3857, 2722} \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {2 n \cos (e+f x) (d \csc (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-2),-\frac {n}{2},\sin ^2(e+f x)\right )}{3 a^2 d^2 f \sqrt {\cos ^2(e+f x)}}-\frac {2 n \cot (e+f x) (d \csc (e+f x))^{n+2}}{3 a^2 d^2 f (\csc (e+f x)+1)}-\frac {(2 n+1) \cos (e+f x) (d \csc (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-n-1),\frac {1-n}{2},\sin ^2(e+f x)\right )}{3 a^2 d f \sqrt {\cos ^2(e+f x)}}+\frac {\cot (e+f x) (d \csc (e+f x))^{n+2}}{3 d^2 f (a \csc (e+f x)+a)^2} \]
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Rule 2722
Rule 3317
Rule 3857
Rule 3872
Rule 3902
Rule 4105
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(d \csc (e+f x))^{2+n}}{(a+a \csc (e+f x))^2} \, dx}{d^2} \\ & = \frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\int \frac {(d \csc (e+f x))^{2+n} (a (-1+n)-a (1+n) \csc (e+f x))}{a+a \csc (e+f x)} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\int (d \csc (e+f x))^{2+n} \left (-a^2 (1+n) (1+2 n)+2 a^2 n (2+n) \csc (e+f x)\right ) \, dx}{3 a^4 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {(2 n (2+n)) \int (d \csc (e+f x))^{3+n} \, dx}{3 a^2 d^3}+\frac {((1+n) (1+2 n)) \int (d \csc (e+f x))^{2+n} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}-\frac {\left (2 n (2+n) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-3-n} \, dx}{3 a^2 d^3}+\frac {\left ((1+n) (1+2 n) (d \csc (e+f x))^n \left (\frac {\sin (e+f x)}{d}\right )^n\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-2-n} \, dx}{3 a^2 d^2} \\ & = -\frac {2 n \cot (e+f x) (d \csc (e+f x))^{2+n}}{3 a^2 d^2 f (1+\csc (e+f x))}+\frac {\cot (e+f x) (d \csc (e+f x))^{2+n}}{3 d^2 f (a+a \csc (e+f x))^2}+\frac {2 n \cot (e+f x) \csc (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2-n),-\frac {n}{2},\sin ^2(e+f x)\right )}{3 a^2 f \sqrt {\cos ^2(e+f x)}}-\frac {(1+2 n) \cos (e+f x) (d \csc (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1-n),\frac {1-n}{2},\sin ^2(e+f x)\right )}{3 a^2 d f \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx \]
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\[\int \frac {\left (d \csc \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]
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\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (d \csc {\left (e + f x \right )}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
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\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (d \csc \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d \csc (e+f x))^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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