Integrand size = 40, antiderivative size = 34 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \]
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Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2573, 6816} \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c} \]
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Rule 2573
Rule 6816
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\sqrt {\frac {1-c x}{1+c x}}\right )\right )} \, dx,\sqrt {\frac {1-c x}{1+c x}},\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right ) \\ & = -\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \]
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\[\int \frac {1}{\left (-x^{2} c^{2}+1\right ) \left (a +b \ln \left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )\right )}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}{b c} \]
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Time = 7.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {\log {\left (x - \frac {1}{c} \right )}}{2 c} + \frac {\log {\left (x + \frac {1}{c} \right )}}{2 c}}{a} & \text {for}\: b = 0 \\\frac {x}{a} & \text {for}\: c = 0 \\- \frac {\log {\left (\frac {a}{b} + \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} \right )}}{b c} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (-\frac {b \log \left (c x + 1\right ) - b \log \left (-c x + 1\right ) - 2 \, a}{2 \, b}\right )}{b c} \]
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Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\frac {\log \left (-b \log \left (c x + 1\right ) + b \log \left (-c x + 1\right ) + 2 \, a\right )}{b c} \]
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Timed out. \[ \int \frac {1}{\left (1-c^2 x^2\right ) \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )} \, dx=-\int \frac {1}{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )\,\left (c^2\,x^2-1\right )} \,d x \]
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