Integrand size = 21, antiderivative size = 21 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a-a \sin (e+f x))},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx \\ \end{align*}
Not integrable
Time = 5.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (d x +c \right ) \left (a -a \sin \left (f x +e \right )\right )}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int { -\frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) - a\right )}} \,d x } \]
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Not integrable
Time = 1.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=- \frac {\int \frac {1}{c \sin {\left (e + f x \right )} - c + d x \sin {\left (e + f x \right )} - d x}\, dx}{a} \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 285, normalized size of antiderivative = 13.57 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int { -\frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) - a\right )}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int { -\frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) - a\right )}} \,d x } \]
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Not integrable
Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a-a \sin (e+f x))} \, dx=\int \frac {1}{\left (a-a\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]
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