Integrand size = 26, antiderivative size = 26 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \\ \end{align*}
Not integrable
Time = 4.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
\[\int \frac {\cos \left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
[In]
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Not integrable
Time = 4.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 0.72 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 2.54 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\cos (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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