Optimal. Leaf size=19 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \]
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Rubi [A] time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3660, 3675, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \]
Antiderivative was successfully verified.
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Rule 206
Rule 3660
Rule 3675
Rubi steps
\begin {align*} \int \frac {1}{1-2 \coth ^2(x)} \, dx &=-x-2 \int \frac {\text {csch}^2(x)}{1-2 \coth ^2(x)} \, dx\\ &=-x+2 \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\coth (x)\right )\\ &=-x+\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 19, normalized size = 1.00 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 70, normalized size = 3.68 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {2} - 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 3}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 38, normalized size = 2.00 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 27, normalized size = 1.42 \[ \frac {\ln \left (\coth \relax (x )-1\right )}{2}-\frac {\ln \left (1+\coth \relax (x )\right )}{2}+\sqrt {2}\, \arctanh \left (\sqrt {2}\, \coth \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 38, normalized size = 2.00 \[ -\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 15, normalized size = 0.79 \[ \sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\mathrm {coth}\relax (x)\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 34, normalized size = 1.79 \[ - x - \frac {\sqrt {2} \log {\left (\tanh {\relax (x )} - \sqrt {2} \right )}}{2} + \frac {\sqrt {2} \log {\left (\tanh {\relax (x )} + \sqrt {2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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