Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \\ \end{align*}
Not integrable
Time = 32.82 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
\[\int \frac {\sec ^{3}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}d x\]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 8.56 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]
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Exception generated. \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Timed out} \]
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Not integrable
Time = 4.50 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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