Optimal. Leaf size=37 \[ -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c} \]
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Rubi [A]
time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2573, 2576, 12,
2338} \begin {gather*} -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2338
Rule 2573
Rule 2576
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 {\text {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*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )}{-c^{2} x^{2}+1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (31) = 62\).
time = 0.29, size = 105, normalized size = 2.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 47, normalized size = 1.27 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (29) = 58\).
time = 4.59, size = 61, normalized size = 1.65 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (31) = 62\).
time = 5.02, size = 86, normalized size = 2.32 \begin {gather*} -\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 {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 {a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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