Optimal. Leaf size=32 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\coth (x)+1} \]
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Rubi [A] time = 0.04, antiderivative size = 32, 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 = {3527, 3480, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\coth (x)+1} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3480
Rule 3527
Rubi steps
\begin {align*} \int \coth (x) \sqrt {1+\coth (x)} \, dx &=-2 \sqrt {1+\coth (x)}+\int \sqrt {1+\coth (x)} \, dx\\ &=-2 \sqrt {1+\coth (x)}+2 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\coth (x)}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 53, normalized size = 1.66 \[ (1+i) \sqrt {\coth (x)+1} \left (-\frac {i \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{\sqrt {i (\coth (x)+1)}}-(1-i)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 131, normalized size = 4.09 \[ -\frac {4 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - {\left (\sqrt {2} \cosh \relax (x)^{2} + 2 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + \sqrt {2} \sinh \relax (x)^{2} - \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{2 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 71, normalized size = 2.22 \[ -\frac {1}{2} \, \sqrt {2} {\left (\log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}{\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.81 \[ \arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\coth \relax (x )} \]
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 \[ \int \sqrt {\coth \relax (x) + 1} \coth \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 25, normalized size = 0.78 \[ \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )-2\,\sqrt {\mathrm {coth}\relax (x)+1} \]
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 \[ \int \sqrt {\coth {\relax (x )} + 1} \coth {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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