Integrand size = 20, antiderivative size = 20 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\text {Int}\left (\frac {(a+a \sec (e+f x))^2}{c+d x},x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx \\ \end{align*}
Not integrable
Time = 30.85 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx \]
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Not integrable
Time = 0.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {\left (a +a \sec \left (f x +e \right )\right )^{2}}{d x +c}d x\]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]
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Not integrable
Time = 1.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{c + d x}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{c + d x}\, dx + \int \frac {1}{c + d x}\, dx\right ) \]
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Not integrable
Time = 1.03 (sec) , antiderivative size = 507, normalized size of antiderivative = 25.35 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]
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Not integrable
Time = 1.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]
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Not integrable
Time = 13.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(a+a \sec (e+f x))^2}{c+d x} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{c+d\,x} \,d x \]
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