Optimal. Leaf size=34 \[ -\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \]
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Rubi [A]
time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2573, 6816}
\begin {gather*} -\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2573
Rule 6816
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 {\text {Subst}\left (\int \frac {1}{x (a+b \log (x))} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {\text {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 = 0.03, size = 34, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-c^{2} x^{2}+1\right ) \left (a +b \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 36, normalized size = 1.06 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 30, normalized size = 0.88 \begin {gather*} -\frac {\log \left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.29, size = 54, normalized size = 1.59 \begin {gather*} \begin {cases} \frac {x}{a} & \text {for}\: c = 0 \wedge \left (b = 0 \vee c = 0\right ) \\\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 {\log {\left (\frac {a}{b} + \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )} \right )}}{b c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.08, size = 31, normalized size = 0.91 \begin {gather*} -\frac {\log \left (-b \log \left (c x + 1\right ) + b \log \left (-c x + 1\right ) + 2 \, a\right )}{b c} \end {gather*}
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 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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