Integrand size = 33, antiderivative size = 33 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\text {Int}\left (\frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)},x\right ) \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx \\ \end{align*}
Not integrable
Time = 6.97 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx \]
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Not integrable
Time = 0.60 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00
\[\int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{a +b \sin \left (f x +e \right )}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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Not integrable
Time = 4.65 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Not integrable
Time = 12.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+b \sin (e+f x)} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
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