Integrand size = 4, antiderivative size = 25 \[ \int \coth ^{-1}(a x) \, dx=x \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{2 a} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6022, 266} \[ \int \coth ^{-1}(a x) \, dx=\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
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Rule 266
Rule 6022
Rubi steps \begin{align*} \text {integral}& = x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2} \, dx \\ & = x \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{2 a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \coth ^{-1}(a x) \, dx=x \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{2 a} \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
method | result | size |
parts | \(x \,\operatorname {arccoth}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a}\) | \(23\) |
derivativedivides | \(\frac {a x \,\operatorname {arccoth}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}-1\right )}{2}}{a}\) | \(25\) |
default | \(\frac {a x \,\operatorname {arccoth}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}-1\right )}{2}}{a}\) | \(25\) |
parallelrisch | \(-\frac {-a x \,\operatorname {arccoth}\left (a x \right )-\ln \left (a x -1\right )-\operatorname {arccoth}\left (a x \right )}{a}\) | \(29\) |
risch | \(\frac {x \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right ) x}{2}+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a}\) | \(35\) |
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none
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \coth ^{-1}(a x) \, dx=\frac {a x \log \left (\frac {a x + 1}{a x - 1}\right ) + \log \left (a^{2} x^{2} - 1\right )}{2 \, a} \]
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Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \coth ^{-1}(a x) \, dx=\begin {cases} x \operatorname {acoth}{\left (a x \right )} + \frac {\log {\left (a x + 1 \right )}}{a} - \frac {\operatorname {acoth}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {i \pi x}{2} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \coth ^{-1}(a x) \, dx=\frac {2 \, a x \operatorname {arcoth}\left (a x\right ) + \log \left (-a^{2} x^{2} + 1\right )}{2 \, a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 6.12 \[ \int \coth ^{-1}(a x) \, dx=a {\left (\frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{2}} - \frac {\log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{2}} + \frac {\log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{2} {\left (\frac {a x + 1}{a x - 1} - 1\right )}}\right )} \]
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Time = 4.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \coth ^{-1}(a x) \, dx=x\,\mathrm {acoth}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2-1\right )}{2\,a} \]
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