Optimal. Leaf size=14 \[ \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4128, 377, 203} \[ \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 377
Rule 4128
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\text {csch}^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {-2+x^2}} \, dx,x,\coth (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.03, size = 48, normalized size = 3.43 \[ \frac {\sqrt {\cosh (2 x)-3} \text {csch}(x) \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)-3}\right )}{\sqrt {2} \sqrt {\text {csch}^2(x)-1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 213, normalized size = 15.21 \[ -\frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 2 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 6 \, {\left (\cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 69, normalized size = 4.93 \[ -\frac {\arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) + 2 \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right )}{2 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-1+\mathrm {csch}\relax (x )^{2}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}\relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {\frac {1}{{\mathrm {sinh}\relax (x)}^2}-1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\operatorname {csch}^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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