Integrand size = 4, antiderivative size = 19 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=x \coth ^{-1}\left (\frac {1}{x}\right )+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6022, 269, 266} \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=\frac {1}{2} \log \left (1-x^2\right )+x \coth ^{-1}\left (\frac {1}{x}\right ) \]
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Rule 266
Rule 269
Rule 6022
Rubi steps \begin{align*} \text {integral}& = x \coth ^{-1}\left (\frac {1}{x}\right )+\int \frac {1}{\left (1-\frac {1}{x^2}\right ) x} \, dx \\ & = x \coth ^{-1}\left (\frac {1}{x}\right )+\int \frac {x}{-1+x^2} \, dx \\ & = x \coth ^{-1}\left (\frac {1}{x}\right )+\frac {1}{2} \log \left (1-x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=x \coth ^{-1}\left (\frac {1}{x}\right )+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(x \,\operatorname {arccoth}\left (\frac {1}{x}\right )+\ln \left (1+x \right )-\operatorname {arccoth}\left (\frac {1}{x}\right )\) | \(18\) |
parts | \(x \,\operatorname {arccoth}\left (\frac {1}{x}\right )+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (1+x \right )}{2}\) | \(20\) |
derivativedivides | \(x \,\operatorname {arccoth}\left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}+1\right )}{2}-\ln \left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}-1\right )}{2}\) | \(30\) |
default | \(x \,\operatorname {arccoth}\left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}+1\right )}{2}-\ln \left (\frac {1}{x}\right )+\frac {\ln \left (\frac {1}{x}-1\right )}{2}\) | \(30\) |
risch | \(-\frac {\ln \left (x -1\right ) x}{2}+\frac {\ln \left (1+x \right ) x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right ) x}{4}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right ) x}{4}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{3} x}{4}+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{3} x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (x -1\right )\right ) \operatorname {csgn}\left (\frac {i \left (x -1\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) x}{4}-\frac {i \pi x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (x -1\right )\right ) \operatorname {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{2} x}{4}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) x}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x}\right )^{2} x}{4}+\frac {\ln \left (x^{2}-1\right )}{2}\) | \(234\) |
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=\frac {1}{2} \, x \log \left (-\frac {x + 1}{x - 1}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \]
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Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=x \operatorname {acoth}{\left (\frac {1}{x} \right )} + \log {\left (x + 1 \right )} - \operatorname {acoth}{\left (\frac {1}{x} \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=x \operatorname {arcoth}\left (\frac {1}{x}\right ) + \frac {1}{2} \, \log \left (x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 5.47 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=\frac {\log \left (-\frac {\frac {\frac {x + 1}{x - 1} + 1}{\frac {x + 1}{x - 1} - 1} + 1}{\frac {\frac {x + 1}{x - 1} + 1}{\frac {x + 1}{x - 1} - 1} - 1}\right )}{\frac {x + 1}{x - 1} - 1} + \log \left (\frac {{\left | -x - 1 \right |}}{{\left | x - 1 \right |}}\right ) - \log \left ({\left | -\frac {x + 1}{x - 1} + 1 \right |}\right ) \]
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Time = 4.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \coth ^{-1}\left (\frac {1}{x}\right ) \, dx=\frac {\ln \left (x^2-1\right )}{2}+x\,\left (\frac {\ln \left (x+1\right )}{2}-\frac {\ln \left (1-x\right )}{2}\right ) \]
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