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 = 0.0071696, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [C] time = 0.178869, 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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Maple [B] time = 0.045, size = 71, normalized size = 1.8 \begin{align*} x-{\frac{1}{2}\ln \left ( x\sqrt{-{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{2}\ln \left ( x\sqrt{-{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{2}\ln \left ( x\sqrt{{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{2}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60909, size = 61, normalized size = 1.52 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.58194, size = 159, normalized size = 3.98 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12146, size = 69, normalized size = 1.72 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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