Optimal. Leaf size=42 \[ -2 \sqrt {\coth (x)+1}-\frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3543, 3479, 3480, 206} \[ -2 \sqrt {\coth (x)+1}-\frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3479
Rule 3480
Rule 3543
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{\sqrt {1+\coth (x)}} \, dx &=-2 \sqrt {1+\coth (x)}+\int \frac {1}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{\sqrt {1+\coth (x)}}-2 \sqrt {1+\coth (x)}+\frac {1}{2} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{\sqrt {1+\coth (x)}}-2 \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)}}-2 \sqrt {1+\coth (x)}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 81, normalized size = 1.93 \[ \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {csch}(x) (\sinh (x)+\cosh (x)) \left (\left (\frac {1}{2}-\frac {i}{2}\right ) (-\sinh (2 x)+\cosh (2 x)-5)-\frac {i \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{\sqrt {i (\coth (x)+1)}}\right )}{\sqrt {\coth (x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 189, normalized size = 4.50 \[ -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} + 10 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + 5 \, \sqrt {2} \sinh \relax (x)^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3} + {\left (3 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x) - \sqrt {2} \cosh \relax (x)\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 88, normalized size = 2.10 \[ -\frac {\frac {5 \, \sqrt {2} e^{\left (2 \, x\right )}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\sqrt {2}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}}{2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}}} - \frac {\sqrt {2} \log \left ({\left | 4 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 4 \, e^{\left (2 \, x\right )} + 2 \right |}\right )}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 35, normalized size = 0.83 \[ \frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\coth \relax (x )}}-2 \sqrt {1+\coth \relax (x )} \]
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 \[ \int \frac {\coth \relax (x)^{2}}{\sqrt {\coth \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 36, normalized size = 0.86 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{2}-\frac {3}{\sqrt {\mathrm {coth}\relax (x)+1}}-\frac {2\,\mathrm {coth}\relax (x)}{\sqrt {\mathrm {coth}\relax (x)+1}} \]
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 \[ \int \frac {\coth ^{2}{\relax (x )}}{\sqrt {\coth {\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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