Optimal. Leaf size=37 \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 b c} \]
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Rubi [A] time = 0.0379821, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2512, 2301} \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 b c} \]
Antiderivative was successfully verified.
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Rule 2512
Rule 2301
Rubi steps
\begin{align*} \int \frac{a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \log (x)}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{2 b c}\\ \end{align*}
Mathematica [A] time = 0.0063933, size = 37, normalized size = 1. \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 b c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.364, 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 ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20672, size = 142, normalized size = 3.84 \begin{align*} \frac{1}{2} \, b{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + \frac{1}{2} \, a{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} + \frac{{\left (\log \left (c x + 1\right )^{2} - 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right ) + \log \left (c x - 1\right )^{2}\right )} b}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99544, size = 119, normalized size = 3.22 \begin{align*} -\frac{b \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.8092, size = 61, normalized size = 1.65 \begin{align*} \begin{cases} - \frac{a \operatorname{atan}{\left (\frac{x}{\sqrt{- \frac{1}{c^{2}}}} \right )}}{c^{2} \sqrt{- \frac{1}{c^{2}}}} & \text{for}\: b = 0 \\a x & \text{for}\: c = 0 \\- \frac{\left (a + b \log{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}\right )^{2}}{2 b c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26335, size = 116, normalized size = 3.14 \begin{align*} -\frac{b \log \left (c x + 1\right )^{2}}{8 \, c} + \frac{b \log \left (c x - 1\right )^{2}}{8 \, c} + \frac{1}{4} \,{\left (\frac{b \log \left (c x + 1\right )}{c} - \frac{b \log \left (c x - 1\right )}{c}\right )} \log \left (-c x + 1\right ) + \frac{a \log \left (c x + 1\right )}{2 \, c} - \frac{a \log \left (c x - 1\right )}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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