Optimal. Leaf size=19 \[ -x+\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.04, 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 = {3741, 3756,
212} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3741
Rule 3756
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 \text {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 = 0.08, size = 19, normalized size = 1.00 \begin {gather*} -x+\sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 27, normalized size = 1.42
method | result | size |
derivativedivides | \(\sqrt {2}\, \arctanh \left (\coth \left (x \right ) \sqrt {2}\right )+\frac {\ln \left (\coth \left (x \right )-1\right )}{2}-\frac {\ln \left (1+\coth \left (x \right )\right )}{2}\) | \(27\) |
default | \(\sqrt {2}\, \arctanh \left (\coth \left (x \right ) \sqrt {2}\right )+\frac {\ln \left (\coth \left (x \right )-1\right )}{2}-\frac {\ln \left (1+\coth \left (x \right )\right )}{2}\) | \(27\) |
risch | \(-x +\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3-2 \sqrt {2}\right )}{2}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{2}\) | \(39\) |
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. 38 vs.
\(2 (16) = 32\).
time = 0.49, size = 38, normalized size = 2.00 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (16) = 32\).
time = 0.39, size = 70, normalized size = 3.68 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 34, normalized size = 1.79 \begin {gather*} - x - \frac {\sqrt {2} \log {\left (\tanh {\left (x \right )} - \sqrt {2} \right )}}{2} + \frac {\sqrt {2} \log {\left (\tanh {\left (x \right )} + \sqrt {2} \right )}}{2} \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. 38 vs.
\(2 (16) = 32\).
time = 0.38, size = 38, normalized size = 2.00 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\mathrm {coth}\left (x\right )\right )-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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