Integrand size = 31, antiderivative size = 60 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)} \]
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Time = 0.09 (sec) , antiderivative size = 60, 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} \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=-\frac {(c+d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]
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Rule 70
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (c+\frac {d x}{a}\right )^n}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = -\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)} \]
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\[\int \frac {\cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{a +a \sin \left (f x +e \right )}d x\]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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