Integrand size = 21, antiderivative size = 149 \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\frac {3 d \cos (e+f x) (d \csc (e+f x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {3 d^2 \cos (e+f x) (d \csc (e+f x))^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3317, 3872, 3857, 2722} \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\frac {a d^2 \cos (e+f x) (d \csc (e+f x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right )}{f (2-n) \sqrt {\cos ^2(e+f x)}}+\frac {a d \cos (e+f x) (d \csc (e+f x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right )}{f (1-n) \sqrt {\cos ^2(e+f x)}} \]
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Rule 2722
Rule 3317
Rule 3857
Rule 3872
Rubi steps \begin{align*} \text {integral}& = d \int (d \csc (e+f x))^{-1+n} (a+a \csc (e+f x)) \, dx \\ & = a \int (d \csc (e+f x))^n \, dx+(a d) \int (d \csc (e+f x))^{-1+n} \, dx \\ & = \left (a (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 )^{-n} \, dx+\left (a d (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 )^{1-n} \, dx \\ & = \frac {a \cos (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\sin ^2(e+f x)\right ) \sin (e+f x)}{f (1-n) \sqrt {\cos ^2(e+f x)}}+\frac {a \cos (e+f x) (d \csc (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2-n}{2},\frac {4-n}{2},\sin ^2(e+f x)\right ) \sin ^2(e+f x)}{f (2-n) \sqrt {\cos ^2(e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.88 \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\frac {3\ 2^{-1+n} e^{-i (e+f n x)} \left (\frac {i e^{i (e+f x)}}{-1+e^{2 i (e+f x)}}\right )^n \left (-1+e^{2 i (e+f x)}\right ) \csc ^{-1-n}(e+f x) (d \csc (e+f x))^n (1+\csc (e+f x)) \left (-e^{i f (-1+n) x} n (1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {1+n}{2},e^{2 i (e+f x)}\right )+e^{i e} (-1+n) \left (e^{i (e+f (1+n) x)} n \operatorname {Hypergeometric2F1}\left (1,\frac {3-n}{2},\frac {3+n}{2},e^{2 i (e+f x)}\right )+2 i e^{i f n x} (1+n) \operatorname {Hypergeometric2F1}\left (1,1-\frac {n}{2},\frac {2+n}{2},e^{2 i (e+f x)}\right )\right )\right )}{f (-1+n) n (1+n) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
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\[\int \left (d \csc \left (f x +e \right )\right )^{n} \left (a +a \sin \left (f x +e \right )\right )d x\]
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\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=a \left (\int \left (d \csc {\left (e + f x \right )}\right )^{n}\, dx + \int \left (d \csc {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx\right ) \]
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\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} \left (d \csc \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (d \csc (e+f x))^n (3+3 \sin (e+f x)) \, dx=\int {\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^n\,\left (a+a\,\sin \left (e+f\,x\right )\right ) \,d x \]
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