Optimal. Leaf size=32 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\coth (x)} \]
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Rubi [A]
time = 0.03, 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 = {3608, 3561,
212} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\coth (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3561
Rule 3608
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 \text {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] Result contains complex when optimal does not.
time = 0.08, size = 53, normalized size = 1.66 \begin {gather*} (1+i) \sqrt {1+\coth (x)} \left ((-1+i)-\frac {i \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right )}{\sqrt {i (1+\coth (x))}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 26, normalized size = 0.81
method | result | size |
derivativedivides | \(\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\coth \left (x \right )}\) | \(26\) |
default | \(\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\coth \left (x \right )}\) | \(26\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (25) = 50\).
time = 0.36, size = 131, normalized size = 4.09 \begin {gather*} -\frac {4 \, \sqrt {2} {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )}{2 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \end {gather*}
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 \begin {gather*} \int \sqrt {\coth {\left (x \right )} + 1} \coth {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (25) = 50\).
time = 0.41, size = 71, normalized size = 2.22 \begin {gather*} -\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 )} \end {gather*}
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 \begin {gather*} \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {coth}\left (x\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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