Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+a \sec (e+f x))},x\right ) \]
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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 = {} \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx \\ \end{align*}
Not integrable
Time = 6.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +a \sec \left (f x +e \right )\right )}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 2.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\frac {\int \frac {1}{c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d x \sec {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sec {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \]
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Not integrable
Time = 0.90 (sec) , antiderivative size = 522, normalized size of antiderivative = 26.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 0.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 13.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx=\int \frac {1}{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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