Optimal. Leaf size=12 \[ \frac {1}{2} \log (\sinh (a+2 \log (x))) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3556}
\begin {gather*} \frac {1}{2} \log (\sinh (a+2 \log (x))) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \frac {\coth (a+2 \log (x))}{x} \, dx &=\text {Subst}(\int \coth (a+2 x) \, dx,x,\log (x))\\ &=\frac {1}{2} \log (\sinh (a+2 \log (x)))\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.75 \begin {gather*} \frac {1}{2} (\log (\cosh (a+2 \log (x)))+\log (\tanh (a+2 \log (x)))) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(25\) vs.
\(2(10)=20\).
time = 0.79, size = 26, normalized size = 2.17
method | result | size |
risch | \(-\ln \left (x \right )+\frac {\ln \left (1-{\mathrm e}^{2 a} x^{4}\right )}{2}\) | \(20\) |
derivativedivides | \(-\frac {\ln \left (\coth \left (a +2 \ln \left (x \right )\right )-1\right )}{4}-\frac {\ln \left (\coth \left (a +2 \ln \left (x \right )\right )+1\right )}{4}\) | \(26\) |
default | \(-\frac {\ln \left (\coth \left (a +2 \ln \left (x \right )\right )-1\right )}{4}-\frac {\ln \left (\coth \left (a +2 \ln \left (x \right )\right )+1\right )}{4}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \log \left (\sinh \left (a + 2 \, \log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (10) = 20\).
time = 0.53, size = 27, normalized size = 2.25 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (\tanh {\left (a + 2 \log {\left (x \right )} \right )} + 1 \right )}}{2} + \frac {\log {\left (\tanh {\left (a + 2 \log {\left (x \right )} \right )} \right )}}{2} \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. 21 vs.
\(2 (10) = 20\).
time = 0.39, size = 21, normalized size = 1.75 \begin {gather*} -\frac {1}{4} \, \log \left (x^{4}\right ) + \frac {1}{2} \, \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 18, normalized size = 1.50 \begin {gather*} \frac {\ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )}{2}-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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