Optimal. Leaf size=41 \[ e^{a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac {1}{x} \]
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Rubi [F] time = 0.02, 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 = {} \[ \int \frac {\coth (a+2 \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\coth (a+2 \log (x))}{x^2} \, dx &=\int \frac {\coth (a+2 \log (x))}{x^2} \, dx\\ \end {align*}
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Mathematica [C] time = 0.17, size = 62, normalized size = 1.51 \[ \frac {x (\sinh (a)+\cosh (a))^2 \text {RootSum}\left [-\text {$\#$1}^4 \sinh (a)+\text {$\#$1}^4 \cosh (a)-\sinh (a)-\cosh (a)\& ,\frac {\log \left (\frac {1}{x}-\text {$\#$1}\right )+\log (x)}{\text {$\#$1}^3}\& \right ]+2}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 54, normalized size = 1.32 \[ \frac {2 \, x \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + x e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) + 2}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 52, normalized size = 1.27 \[ \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 93, normalized size = 2.27 \[ \frac {1}{x}+\frac {\sqrt {-{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2}-\frac {\sqrt {-{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x -\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2}+\frac {\sqrt {{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x +\left ({\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2}-\frac {\sqrt {{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x -\left ({\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 47, normalized size = 1.15 \[ -\arctan \left (\frac {e^{\left (-\frac {1}{2} \, a\right )}}{x}\right ) e^{\left (\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\frac {1}{x} - e^{\left (\frac {1}{2} \, a\right )}}{\frac {1}{x} + e^{\left (\frac {1}{2} \, a\right )}}\right ) + \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 37, normalized size = 0.90 \[ {\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )-{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )+\frac {1}{x} \]
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 \[ \int \frac {\coth {\left (a + 2 \log {\relax (x )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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