Optimal. Leaf size=7 \[ -\log (1+\coth (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3568, 31}
\begin {gather*} -\log (\coth (x)+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3568
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{1+\coth (x)} \, dx &=-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -x+\log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 8, normalized size = 1.14
method | result | size |
derivativedivides | \(-\ln \left (1+\coth \left (x \right )\right )\) | \(8\) |
default | \(-\ln \left (1+\coth \left (x \right )\right )\) | \(8\) |
risch | \(-2 x +\ln \left ({\mathrm e}^{2 x}-1\right )\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 7, normalized size = 1.00 \begin {gather*} -\log \left (\coth \left (x\right ) + 1\right ) \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. 18 vs.
\(2 (7) = 14\).
time = 0.42, size = 18, normalized size = 2.57 \begin {gather*} -2 \, x + \log \left (\frac {2 \, \sinh \left (x\right )}{\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 {\operatorname {csch}^{2}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 12, normalized size = 1.71 \begin {gather*} -2 \, x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 11, normalized size = 1.57 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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