Integrand size = 34, antiderivative size = 34 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\text {Int}\left (\frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{(1-a x) (1+a x)},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sec (x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ \end{align*}
Not integrable
Time = 3.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
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Not integrable
Time = 0.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
\[\int \frac {1}{\left (-a^{2} x^{2}+1\right ) \cos \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]
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Not integrable
Time = 6.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {1}{a^{2} x^{2} \cos {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \cos {\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]
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Not integrable
Time = 0.47 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]
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Not integrable
Time = 0.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cos \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )} \,d x } \]
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Not integrable
Time = 26.47 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\sec \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {1}{\cos \left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )\,\left (a^2\,x^2-1\right )} \,d x \]
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