Integrand size = 20, antiderivative size = 20 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\text {Int}\left (\frac {(e+f x)^m}{a+b \sin (c+d x)},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx \\ \end{align*}
Not integrable
Time = 0.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x +e \right )^{m}}{a +b \sin \left (d x +c \right )}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 0.94 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{m}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 0.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 2.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^m}{a+b\,\sin \left (c+d\,x\right )} \,d x \]
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