Optimal. Leaf size=40 \[ x-e^{-a/2} \text {ArcTan}\left (e^{a/2} x\right )-e^{-a/2} \tanh ^{-1}\left (e^{a/2} x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5653, 396, 218,
212, 209} \begin {gather*} -e^{-a/2} \text {ArcTan}\left (e^{a/2} x\right )-e^{-a/2} \tanh ^{-1}\left (e^{a/2} x\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 396
Rule 5653
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] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.13, size = 58, normalized size = 1.45 \begin {gather*} x+\frac {1}{2} \text {RootSum}\left [-\cosh (a)+\sinh (a)+\cosh (a) \text {$\#$1}^4+\sinh (a) \text {$\#$1}^4\&,\frac {\log (x)-\log (x-\text {$\#$1})}{\text {$\#$1}^3}\&\right ] (-\cosh (2 a)+\sinh (2 a)) \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. \(70\) vs.
\(2(32)=64\).
time = 0.71, size = 71, normalized size = 1.78
method | result | size |
risch | \(x +\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x -1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}+1\right )}{2 \sqrt {-{\mathrm e}^{a}}}+\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}-1\right )}{2 \sqrt {-{\mathrm e}^{a}}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 45, normalized size = 1.12 \begin {gather*} -\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 {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. 58 vs.
\(2 (28) = 56\).
time = 0.36, size = 58, normalized size = 1.45 \begin {gather*} -\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 {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 \coth {\left (a + 2 \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 51, normalized size = 1.28 \begin {gather*} -\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 {gather*}
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 \begin {gather*} 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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