Integrand size = 10, antiderivative size = 19 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=-x+\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3741, 3756, 212} \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )-x \]
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Rule 212
Rule 3741
Rule 3756
Rubi steps \begin{align*} \text {integral}& = -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} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=-x+\sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(-\frac {\ln \left (1+\coth \left (x \right )\right )}{2}+\frac {\ln \left (\coth \left (x \right )-1\right )}{2}+\sqrt {2}\, \operatorname {arctanh}\left (\coth \left (x \right ) \sqrt {2}\right )\) | \(27\) |
default | \(-\frac {\ln \left (1+\coth \left (x \right )\right )}{2}+\frac {\ln \left (\coth \left (x \right )-1\right )}{2}+\sqrt {2}\, \operatorname {arctanh}\left (\coth \left (x \right ) \sqrt {2}\right )\) | \(27\) |
risch | \(-x +\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3-2 \sqrt {2}\right )}{2}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{2}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (16) = 32\).
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) - x \]
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=- x - \frac {\sqrt {2} \log {\left (\tanh {\left (x \right )} - \sqrt {2} \right )}}{2} + \frac {\sqrt {2} \log {\left (\tanh {\left (x \right )} + \sqrt {2} \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) - x \]
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Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{1-2 \coth ^2(x)} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\mathrm {coth}\left (x\right )\right )-x \]
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