Optimal. Leaf size=31 \[ -\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1180, 213}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1180
Rubi steps
\begin {align*} \int \frac {-5+2 x^2}{6-5 x^2+x^4} \, dx &=\int \frac {1}{-3+x^2} \, dx+\int \frac {1}{-2+x^2} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(31)=62\).
time = 0.01, size = 69, normalized size = 2.23 \begin {gather*} \frac {1}{12} \left (3 \sqrt {2} \log \left (\sqrt {2}-x\right )+2 \sqrt {3} \log \left (\sqrt {3}-x\right )-3 \sqrt {2} \log \left (\sqrt {2}+x\right )-2 \sqrt {3} \log \left (\sqrt {3}+x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 26, normalized size = 0.84
method | result | size |
default | \(-\frac {\arctanh \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\arctanh \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(26\) |
risch | \(\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right )}{4}+\frac {\sqrt {3}\, \ln \left (x -\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (x +\sqrt {3}\right )}{6}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.72, size = 43, normalized size = 1.39 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) \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. 51 vs.
\(2 (25) = 50\).
time = 1.02, size = 51, normalized size = 1.65 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{2} - 2 \, \sqrt {2} x + 2}{x^{2} - 2}\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 60, normalized size = 1.94 \begin {gather*} \frac {\sqrt {2} \log {\left (x - \sqrt {2} \right )}}{4} - \frac {\sqrt {2} \log {\left (x + \sqrt {2} \right )}}{4} + \frac {\sqrt {3} \log {\left (x - \sqrt {3} \right )}}{6} - \frac {\sqrt {3} \log {\left (x + \sqrt {3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (25) = 50\).
time = 0.48, size = 59, normalized size = 1.90 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 25, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,x}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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