Optimal. Leaf size=26 \[ \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\sin ^{-1}\left (\frac {\coth (x)}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4128, 402, 216, 377, 206} \[ \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\sin ^{-1}\left (\frac {\coth (x)}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 216
Rule 377
Rule 402
Rule 4128
Rubi steps
\begin {align*} \int \sqrt {1-\text {csch}^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {2-x^2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2}} \, dx,x,\coth (x)\right )+\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2}} \, dx,x,\coth (x)\right )\\ &=\sin ^{-1}\left (\frac {\coth (x)}{\sqrt {2}}\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )\\ &=\sin ^{-1}\left (\frac {\coth (x)}{\sqrt {2}}\right )+\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.07, size = 65, normalized size = 2.50 \[ \frac {\sinh (x) \sqrt {2-2 \text {csch}^2(x)} \left (\log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)-3}\right )+\tan ^{-1}\left (\frac {\sqrt {2} \cosh (x)}{\sqrt {\cosh (2 x)-3}}\right )\right )}{\sqrt {\cosh (2 x)-3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 221, normalized size = 8.50 \[ -2 \, \arctan \left (-\frac {1}{2} \, \cosh \relax (x)^{2} - \cosh \relax (x) \sinh \relax (x) - \frac {1}{2} \, \sinh \relax (x)^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 2\right )} \sinh \relax (x)^{2} - \sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 2 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1\right ) + \frac {1}{2} \, \log \left (-\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) - \sinh \relax (x)^{2} + \sqrt {2} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 110, normalized size = 4.23 \[ -\frac {1}{2} \, {\left (4 \, \arctan \left (\frac {1}{2} \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - \frac {1}{2} \, e^{\left (2 \, x\right )} + \frac {1}{2}\right ) + \log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) + \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) - \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \sqrt {1-\mathrm {csch}\relax (x )^{2}}\, dx \]
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 \[ \int \sqrt {-\operatorname {csch}\relax (x)^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {1-\frac {1}{{\mathrm {sinh}\relax (x)}^2}} \,d x \]
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 \[ \int \sqrt {1 - \operatorname {csch}^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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