Integrand size = 36, antiderivative size = 36 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\text {Int}\left (\frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{(1-a x) (1+a x)},x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 36, 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 = {} \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\text {sech}^2(x)}{x} \, dx,x,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a} \\ \end{align*}
Not integrable
Time = 32.42 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
\[\int \frac {1}{\left (-a^{2} x^{2}+1\right ) \cosh \left (\frac {\sqrt {-a x +1}}{\sqrt {a x +1}}\right )^{2}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]
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Not integrable
Time = 22.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=- \int \frac {1}{a^{2} x^{2} \cosh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )} - \cosh ^{2}{\left (\frac {\sqrt {- a x + 1}}{\sqrt {a x + 1}} \right )}}\, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.33 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=\int { -\frac {1}{{\left (a^{2} x^{2} - 1\right )} \cosh \left (\frac {\sqrt {-a x + 1}}{\sqrt {a x + 1}}\right )^{2}} \,d x } \]
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Not integrable
Time = 1.87 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\text {sech}^2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )}{1-a^2 x^2} \, dx=-\int \frac {1}{{\mathrm {cosh}\left (\frac {\sqrt {1-a\,x}}{\sqrt {a\,x+1}}\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]
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