Optimal. Leaf size=34 \[ -\frac{\log \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{b c} \]
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Rubi [A] time = 0.0677073, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2512, 2302, 29} \[ -\frac{\log \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 2512
Rule 2302
Rule 29
Rubi steps
\begin{align*} \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{\operatorname{Subst}\left (\int \frac{1}{x (a+b \log (x))} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ &=-\frac{\log \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ \end{align*}
Mathematica [A] time = 0.0360779, size = 34, normalized size = 1. \[ -\frac{\log \left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{b c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.351, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b\ln \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48186, size = 49, normalized size = 1.44 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00456, size = 72, normalized size = 2.12 \begin{align*} -\frac{\log \left (b \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 164.757, size = 53, normalized size = 1.56 \begin{align*} \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} + \frac{\log{\left (- c x + 1 \right )}}{2} - \frac{\log{\left (c x + 1 \right )}}{2} \right )}}{b c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16082, size = 42, normalized size = 1.24 \begin{align*} -\frac{\log \left (-b \log \left (c x + 1\right ) + b \log \left (-c x + 1\right ) + 2 \, a\right )}{b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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