Integrand size = 28, antiderivative size = 28 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Int}\left (\frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx \\ \end{align*}
Not integrable
Time = 15.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx \]
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Not integrable
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x +e \right )^{m} \left (\sec ^{2}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 79.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\left (e + f x\right )^{m} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Exception generated. \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 3.86 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^m}{{\cos \left (c+d\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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