Integrand size = 31, antiderivative size = 63 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=-\frac {d^2 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a^3 (c-d)^3 f (1+n)} \]
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Time = 0.09 (sec) , antiderivative size = 63, 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} \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=-\frac {d^2 (c+d \sin (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (3,n+1,n+2,\frac {c+d \sin (e+f x)}{c-d}\right )}{a^3 f (n+1) (c-d)^3} \]
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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)^3} \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = -\frac {d^2 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a^3 (c-d)^3 f (1+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\frac {d^2 \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,-\frac {c+d \sin (e+f x)}{-c+d}\right ) (c+d \sin (e+f x))^{1+n}}{a^3 (-c+d)^3 f (1+n)} \]
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\[\int \frac {\cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{3}}d x\]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^3} \, dx=\int \frac {\cos \left (e+f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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