Optimal. Leaf size=30 \[ \frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3526, 3480, 206} \[ \frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3480
Rule 3526
Rubi steps
\begin {align*} \int \frac {\coth (x)}{\sqrt {1+\coth (x)}} \, dx &=\frac {1}{\sqrt {1+\coth (x)}}+\frac {1}{2} \int \sqrt {1+\coth (x)} \, dx\\ &=\frac {1}{\sqrt {1+\coth (x)}}+\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\coth (x)}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 97, normalized size = 3.23 \[ \frac {\text {csch}(x) \left (\frac {1}{2} \sinh (2 x)-\frac {1}{2} \cosh (2 x)+\frac {1}{2}\right ) (\sinh (x)+\cosh (x))}{\sqrt {\coth (x)+1}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {csch}(x) (\sinh (x)+\cosh (x)) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i \coth (x)+i}\right )}{\sqrt {i \coth (x)+i} \sqrt {\coth (x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 85, normalized size = 2.83 \[ \frac {{\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right ) + 4 \, \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}}}{4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 88, normalized size = 2.93 \[ -\frac {1}{2} \, \sqrt {2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {\sqrt {2} \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\sqrt {2}}{2 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 25, normalized size = 0.83 \[ \frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\coth \relax (x )}} \]
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 \[ \int \frac {\coth \relax (x)}{\sqrt {\coth \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 24, normalized size = 0.80 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{2}+\frac {1}{\sqrt {\mathrm {coth}\relax (x)+1}} \]
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 \[ \int \frac {\coth {\relax (x )}}{\sqrt {\coth {\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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