Integrand size = 12, antiderivative size = 27 \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 27, 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 = {3742, 385, 209} \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )}{\sqrt {2}} \]
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Rule 209
Rule 385
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {1-\coth ^2(x)}}\right ) \sqrt {1+\coth ^2(x)}}{\sqrt {2} \sqrt {-1-\coth ^2(x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(22)=44\).
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}\) | \(66\) |
default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}\) | \(66\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 6.48 \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\frac {1}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (i \, \sqrt {2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (-i \, \sqrt {2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + i \, \sqrt {2} e^{\left (4 \, x\right )} + i \, \sqrt {2} e^{\left (2 \, x\right )} + 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) + \frac {1}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - i \, \sqrt {2} e^{\left (4 \, x\right )} - i \, \sqrt {2} e^{\left (2 \, x\right )} - 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) \]
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\[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\int \frac {1}{\sqrt {- \coth ^{2}{\left (x \right )} - 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\int { \frac {1}{\sqrt {-\coth \left (x\right )^{2} - 1}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=-\frac {\sqrt {2} {\left (-i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )}}{4 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]
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Time = 2.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {coth}\left (x\right )}{\sqrt {-{\mathrm {coth}\left (x\right )}^2-1}}\right )}{2} \]
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