Integrand size = 12, antiderivative size = 13 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3738, 4207, 197} \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Rule 197
Rule 3738
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx \\ & = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right ) \\ & = \frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]
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Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\coth \left (x \right )}{\sqrt {1-\coth \left (x \right )^{2}}}\) | \(14\) |
default | \(\frac {\coth \left (x \right )}{\sqrt {1-\coth \left (x \right )^{2}}}\) | \(14\) |
risch | \(\frac {{\mathrm e}^{2 x}}{2 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {1}{2 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.08 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sqrt {-\frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{\left (-x\right )} \]
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\[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\int \frac {1}{\sqrt {1 - \coth ^{2}{\left (x \right )}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]
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Time = 2.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {-\frac {1}{{\mathrm {cosh}\left (x\right )}^2-1}} \]
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