Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{\sqrt {1+\coth (x)}} \]
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Rubi [A]
time = 0.03, 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 = {3607, 3561,
212} \begin {gather*} \frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 3561
Rule 3607
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)}}+\text {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] Result contains complex when optimal does not.
time = 0.14, size = 97, normalized size = 3.23 \begin {gather*} \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i+i \coth (x)}\right ) \text {csch}(x) (\cosh (x)+\sinh (x))}{\sqrt {i+i \coth (x)} \sqrt {1+\coth (x)}}+\frac {\text {csch}(x) (\cosh (x)+\sinh (x)) \left (\frac {1}{2}-\frac {1}{2} \cosh (2 x)+\frac {1}{2} \sinh (2 x)\right )}{\sqrt {1+\coth (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 25, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\coth \left (x \right )}}\) | \(25\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}+\frac {1}{\sqrt {1+\coth \left (x \right )}}\) | \(25\) |
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. 85 vs.
\(2 (24) = 48\).
time = 0.35, size = 85, normalized size = 2.83 \begin {gather*} \frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\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 ) + 4 \, \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\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 \frac {\coth {\left (x \right )}}{\sqrt {\coth {\left (x \right )} + 1}}\, 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. 64 vs.
\(2 (24) = 48\).
time = 0.41, size = 64, normalized size = 2.13 \begin {gather*} -\frac {\sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} + \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{4 \, \mathrm {sgn}\left (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 = 1.23, size = 24, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{2}+\frac {1}{\sqrt {\mathrm {coth}\left (x\right )+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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