Optimal. Leaf size=45 \[ \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3661, 402, 217, 203, 377} \[ \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 377
Rule 402
Rule 3661
Rubi steps
\begin {align*} \int \sqrt {-1-\coth ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {-1-x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\coth (x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 1.38 \[ \frac {\sinh (x) \sqrt {-\coth ^2(x)-1} \left (\sqrt {2} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )-\tanh ^{-1}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right )\right )}{\sqrt {\cosh (2 x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.46, size = 228, normalized size = 5.07 \[ -\frac {1}{4} \, \sqrt {-2} \log \left (-{\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left ({\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left (-2 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + \sqrt {-2} e^{\left (4 \, x\right )} + \sqrt {-2} e^{\left (2 \, x\right )} + 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) - \frac {1}{4} \, \sqrt {-2} \log \left (-2 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - \sqrt {-2} e^{\left (4 \, x\right )} - \sqrt {-2} e^{\left (2 \, x\right )} - 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) - \frac {1}{2} i \, \log \left ({\left (4 i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) + \frac {1}{2} i \, \log \left ({\left (-4 i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 4 \, e^{\left (2 \, x\right )} - 4\right )} e^{\left (-2 \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.20, size = 124, normalized size = 2.76 \[ -\frac {1}{2} \, \sqrt {2} {\left (i \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) + i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) - i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 142, normalized size = 3.16 \[ -\frac {\sqrt {-\left (\coth \relax (x )-1\right )^{2}-2 \coth \relax (x )}}{2}+\frac {\arctan \left (\frac {\coth \relax (x )}{\sqrt {-\left (\coth \relax (x )-1\right )^{2}-2 \coth \relax (x )}}\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \relax (x )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \relax (x )-1\right )^{2}-2 \coth \relax (x )}}\right )}{2}+\frac {\sqrt {-\left (1+\coth \relax (x )\right )^{2}+2 \coth \relax (x )}}{2}+\frac {\arctan \left (\frac {\coth \relax (x )}{\sqrt {-\left (1+\coth \relax (x )\right )^{2}+2 \coth \relax (x )}}\right )}{2}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \relax (x )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \relax (x )\right )^{2}+2 \coth \relax (x )}}\right )}{2} \]
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 \[ \int \sqrt {-\coth \relax (x)^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 43, normalized size = 0.96 \[ -\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\mathrm {coth}\relax (x)}{\sqrt {-{\mathrm {coth}\relax (x)}^2-1}}\right )-\ln \left (\mathrm {coth}\relax (x)-\sqrt {-{\mathrm {coth}\relax (x)}^2-1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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 \[ \int \sqrt {- \coth ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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