Integrand size = 31, antiderivative size = 59 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d)^2 f (1+m)} \]
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Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2912, 70} \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\frac {(a \sin (e+f x)+a)^{m+1} \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^2} \]
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Rule 70
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+x)^m}{\left (c+\frac {d x}{a}\right )^2} \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d)^2 f (1+m)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d)^2 f (1+m)} \]
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\[\int \frac {\cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{2}}d x\]
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\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^2} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
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