Integrand size = 25, antiderivative size = 25 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {Int}\left ((3+b \sin (e+f x))^m (c+d \sin (e+f x))^n,x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 25, 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 = {} \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \\ \end{align*}
Not integrable
Time = 2.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
\[\int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Exception generated. \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Not integrable
Time = 2.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Not integrable
Time = 0.91 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]
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Not integrable
Time = 11.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
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