Optimal. Leaf size=27 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3661, 377, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 377
Rule 3661
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-\coth ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.70 \[ \frac {\sqrt {\cosh (2 x)} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )}{\sqrt {2} \sqrt {-\coth ^2(x)-1}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 175, normalized size = 6.48 \[ \frac {1}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (i \, \sqrt {2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{8} i \, \sqrt {2} \log \left (\frac {1}{2} \, {\left (-i \, \sqrt {2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - \frac {1}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + i \, \sqrt {2} e^{\left (4 \, x\right )} + i \, \sqrt {2} e^{\left (2 \, x\right )} + 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) + \frac {1}{8} i \, \sqrt {2} \log \left ({\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - i \, \sqrt {2} e^{\left (4 \, x\right )} - i \, \sqrt {2} e^{\left (2 \, x\right )} - 2 i \, \sqrt {2}\right )} e^{\left (-4 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.16, size = 73, normalized size = 2.70 \[ -\frac {\sqrt {2} {\left (-i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) + i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )}}{4 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 66, normalized size = 2.44 \[ -\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \relax (x )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \relax (x )-1\right )^{2}-2 \coth \relax (x )}}\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \relax (x )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \relax (x )\right )^{2}+2 \coth \relax (x )}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-\coth \relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 22, normalized size = 0.81 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {coth}\relax (x)}{\sqrt {-{\mathrm {coth}\relax (x)}^2-1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \coth ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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