Integrand size = 28, antiderivative size = 28 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Int}\left (\frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)},x\right ) \]
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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 = {} \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \\ \end{align*}
Not integrable
Time = 16.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
\[\int \frac {\left (f x +e \right )^{m} \left (\sec ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 72.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{m} \sec ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Not integrable
Time = 19.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Not integrable
Time = 5.54 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^m}{{\cos \left (c+d\,x\right )}^2\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \]
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