Optimal. Leaf size=37 \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{3 b c} \]
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Rubi [A] time = 0.0628152, antiderivative size = 37, 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, 30} \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{3 b c} \]
Antiderivative was successfully verified.
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Rule 2512
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b \log (x))^2}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ &=-\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3}{3 b c}\\ \end{align*}
Mathematica [A] time = 0.0100674, size = 37, normalized size = 1. \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{3 b c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.342, 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 ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27657, size = 362, normalized size = 9.78 \begin{align*} \frac{1}{2} \, b^{2}{\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 )^{2} + a 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}{24} \, b^{2}{\left (\frac{6 \,{\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 )} \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )}{c} + \frac{\log \left (c x + 1\right )^{3} - 3 \, \log \left (c x + 1\right )^{2} \log \left (c x - 1\right ) + 3 \, \log \left (c x + 1\right ) \log \left (c x - 1\right )^{2} - \log \left (c x - 1\right )^{3}}{c}\right )} + \frac{1}{2} \, a^{2}{\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 )} a b}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12897, size = 184, normalized size = 4.97 \begin{align*} -\frac{b^{2} \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 23.3409, size = 65, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{a^{2} \operatorname{atan}{\left (\frac{x}{\sqrt{- \frac{1}{c^{2}}}} \right )}}{c^{2} \sqrt{- \frac{1}{c^{2}}}} & \text{for}\: b = 0 \\a^{2} x & \text{for}\: c = 0 \\- \frac{\left (a + b \log{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}\right )^{3}}{3 b c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b \log \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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