Integrand size = 8, antiderivative size = 32 \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\coth (x)}} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3560, 3561, 212} \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {\coth (x)+1}} \]
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Rule 212
Rule 3560
Rule 3561
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{\sqrt {1+\coth (x)}}+\frac {1}{2} \int \sqrt {1+\coth (x)} \, dx \\ & = -\frac {1}{\sqrt {1+\coth (x)}}+\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right ) \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\coth (x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+\coth (x))\right )}{\sqrt {1+\coth (x)}} \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\coth \left (x \right )}}\) | \(27\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\coth \left (x \right )}}\) | \(27\) |
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.66 \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right ) - 4 \, \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
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\[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\int \frac {1}{\sqrt {\coth {\left (x \right )} + 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\int { \frac {1}{\sqrt {\coth \left (x\right ) + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\frac {\sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} - \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
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Time = 1.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {1+\coth (x)}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{2}-\frac {1}{\sqrt {\mathrm {coth}\left (x\right )+1}} \]
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