Integrand size = 25, antiderivative size = 42 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 67} \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\frac {(a \sin (c+d x)+a)^{m+1} \operatorname {Hypergeometric2F1}(2,m+1,m+2,\sin (c+d x)+1)}{a d (m+1)} \]
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Rule 12
Rule 67
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^2 (a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a \text {Subst}\left (\int \frac {(a+x)^m}{x^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\operatorname {Hypergeometric2F1}(2,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]
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\[\int \cos \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]
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\[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \]
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\[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^m \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^2} \,d x \]
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