Optimal. Leaf size=147 \[ \frac {1}{x}-\frac {e^{a/2} \text {ArcTan}\left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}+\frac {e^{a/2} \text {ArcTan}\left (1+\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}+\frac {e^{a/2} \log \left (1-\sqrt {2} e^{a/2} x+e^a x^2\right )}{2 \sqrt {2}}-\frac {e^{a/2} \log \left (1+\sqrt {2} e^{a/2} x+e^a x^2\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5656, 464,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {e^{a/2} \text {ArcTan}\left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}+\frac {e^{a/2} \text {ArcTan}\left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}+\frac {e^{a/2} \log \left (e^a x^2-\sqrt {2} e^{a/2} x+1\right )}{2 \sqrt {2}}-\frac {e^{a/2} \log \left (e^a x^2+\sqrt {2} e^{a/2} x+1\right )}{2 \sqrt {2}}+\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 464
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5656
Rubi steps
\begin {align*} \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx &=\int \frac {\tanh (a+2 \log (x))}{x^2} \, 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 = 59, normalized size = 0.40 \begin {gather*} \frac {2-x \text {RootSum}\left [\cosh (a)+\sinh (a)+\cosh (a) \text {$\#$1}^4-\sinh (a) \text {$\#$1}^4\&,\frac {\log (x)+\log \left (\frac {1}{x}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ] (\cosh (a)+\sinh (a))^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.61, size = 42, normalized size = 0.29
method | result | size |
risch | \(\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+{\mathrm e}^{2 a}\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4}+4 \,{\mathrm e}^{2 a}\right ) x -\textit {\_R}^{3}\right )\right )}{2}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 125, normalized size = 0.85 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} + \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} - \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{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. 204 vs.
\(2 (101) = 202\).
time = 0.38, size = 204, normalized size = 1.39 \begin {gather*} -\frac {4 \, \sqrt {2} x \arctan \left (-{\left (\sqrt {2} x e^{\left (\frac {5}{2} \, a\right )} - \sqrt {2} \sqrt {x^{2} e^{\left (4 \, a\right )} + \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac {1}{2} \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 4 \, \sqrt {2} x \arctan \left (-{\left (\sqrt {2} x e^{\left (\frac {5}{2} \, a\right )} - \sqrt {2} \sqrt {x^{2} e^{\left (4 \, a\right )} - \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac {1}{2} \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} + \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} - \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - 4}{4 \, x} \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 \frac {\tanh {\left (a + 2 \log {\left (x \right )} \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 121, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 45, normalized size = 0.31 \begin {gather*} \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 )\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}+\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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