Optimal. Leaf size=40 \[ -e^{-a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{-a/2} \tanh ^{-1}\left (e^{a/2} x\right )+x \]
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Rubi [F] time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \coth (a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \coth (a+2 \log (x)) \, dx &=\int \coth (a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [C] time = 0.19, size = 58, normalized size = 1.45 \[ \frac {1}{2} (\sinh (2 a)-\cosh (2 a)) \text {RootSum}\left [\text {$\#$1}^4 \sinh (a)+\text {$\#$1}^4 \cosh (a)+\sinh (a)-\cosh (a)\& ,\frac {\log (x)-\log (x-\text {$\#$1})}{\text {$\#$1}^3}\& \right ]+x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 58, normalized size = 1.45 \[ -\frac {1}{2} \, {\left (2 \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, x e^{a} - e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )\right )} e^{\left (-a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 51, normalized size = 1.28 \[ -\arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 71, normalized size = 1.78 \[ x +\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x -1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}+1\right )}{2 \sqrt {-{\mathrm e}^{a}}}+\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}-1\right )}{2 \sqrt {-{\mathrm e}^{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 45, normalized size = 1.12 \[ -\arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) + x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 36, normalized size = 0.90 \[ x-\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}}-\frac {\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}} \]
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 \[ \int \coth {\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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