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 = 0.07, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 29
Rule 2302
Rule 2512
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*}
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Mathematica [A] time = 0.04, size = 34, normalized size = 1.00 \[ -\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 30, normalized size = 0.88 \[ -\frac {\log \left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 31, normalized size = 0.91 \[ -\frac {\log \left (-b \log \left (c x + 1\right ) + b \log \left (-c x + 1\right ) + 2 \, a\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (b \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )+a \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.76, size = 36, normalized size = 1.06 \[ -\frac {\log \left (-\frac {b \log \left (c x + 1\right ) - b \log \left (-c x + 1\right ) - 2 \, a}{2 \, b}\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {1}{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )\,\left (c^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.96, size = 53, normalized size = 1.56 \[ \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {\log {\left (x - \frac {1}{c} \right )}}{2 c} + \frac {\log {\left (x + \frac {1}{c} \right )}}{2 c}}{a} & \text {for}\: b = 0 \\\frac {x}{a} & \text {for}\: c = 0 \\- \frac {\log {\left (\frac {a}{b} + \frac {\log {\left (- c x + 1 \right )}}{2} - \frac {\log {\left (c x + 1 \right )}}{2} \right )}}{b c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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