Integrand size = 18, antiderivative size = 18 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {a+a \sec (e+f x)}{(c+d x)^2},x\right ) \]
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Not integrable
Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx \\ \end{align*}
Not integrable
Time = 5.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx \]
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Not integrable
Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int \frac {a +a \sec \left (f x +e \right )}{\left (d x +c \right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {a \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 2.55 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=a \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 172, normalized size of antiderivative = 9.56 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {a \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 2.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {a \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]
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Not integrable
Time = 13.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {a+a \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+\frac {a}{\cos \left (e+f\,x\right )}}{{\left (c+d\,x\right )}^2} \,d x \]
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