Optimal. Leaf size=44 \[ -\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1677, 1660,
814} \begin {gather*} -\frac {9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )-\frac {12 x^2+11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 814
Rule 1660
Rule 1677
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {4+x+3 x^2+5 x^3}{x \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {-2+7 x}{x \left (2+3 x+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {9}{1+x}-\frac {8}{2+x}\right ) \, dx,x,x^2\right )\\ &=-\frac {11+12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 44, normalized size = 1.00 \begin {gather*} \frac {-11-12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac {9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 38, normalized size = 0.86
method | result | size |
default | \(\frac {1}{2 x^{2}+2}-\frac {9 \ln \left (x^{2}+1\right )}{2}+\ln \left (x \right )+4 \ln \left (x^{2}+2\right )-\frac {13}{2 \left (x^{2}+2\right )}\) | \(38\) |
norman | \(\frac {-6 x^{2}-\frac {11}{2}}{x^{4}+3 x^{2}+2}-\frac {9 \ln \left (x^{2}+1\right )}{2}+4 \ln \left (x^{2}+2\right )+\ln \left (x \right )\) | \(40\) |
risch | \(\frac {-6 x^{2}-\frac {11}{2}}{x^{4}+3 x^{2}+2}-\frac {9 \ln \left (x^{2}+1\right )}{2}+4 \ln \left (x^{2}+2\right )+\ln \left (x \right )\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 44, normalized size = 1.00 \begin {gather*} -\frac {12 \, x^{2} + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 71, normalized size = 1.61 \begin {gather*} -\frac {12 \, x^{2} - 8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 9 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x\right ) + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 41, normalized size = 0.93 \begin {gather*} \frac {- 12 x^{2} - 11}{2 x^{4} + 6 x^{2} + 4} + \log {\left (x \right )} - \frac {9 \log {\left (x^{2} + 1 \right )}}{2} + 4 \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.17, size = 47, normalized size = 1.07 \begin {gather*} \frac {x^{4} - 21 \, x^{2} - 20}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 40, normalized size = 0.91 \begin {gather*} 4\,\ln \left (x^2+2\right )-\frac {9\,\ln \left (x^2+1\right )}{2}+\ln \left (x\right )-\frac {6\,x^2+\frac {11}{2}}{x^4+3\,x^2+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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