Integrand size = 18, antiderivative size = 18 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\text {Int}\left (\frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x},x\right ) \]
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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 = {} \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx \\ \end{align*}
Not integrable
Time = 29.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int \frac {{\left (a +b \sec \left (d \,x^{2}+c \right )\right )}^{2}}{x}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]
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Not integrable
Time = 2.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}{x}\, dx \]
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Not integrable
Time = 0.49 (sec) , antiderivative size = 286, normalized size of antiderivative = 15.89 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}}{x} \,d x } \]
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Not integrable
Time = 13.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b \sec \left (c+d x^2\right )\right )^2}{x} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2}{x} \,d x \]
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