Optimal. Leaf size=44 \[ -\frac {9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )-\frac {12 x^2+11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]
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Rubi [A] time = 0.08, 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 = {1663, 1646, 800} \[ -\frac {12 x^2+11}{2 \left (x^4+3 x^2+2\right )}-\frac {9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 800
Rule 1646
Rule 1663
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} \operatorname {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} \operatorname {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} \operatorname {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 \[ -\frac {9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\frac {-12 x^2-11}{2 \left (x^4+3 x^2+2\right )}+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 71, normalized size = 1.61 \[ -\frac {12 \, x^{2} - 8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 9 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \relax (x) + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 47, normalized size = 1.07 \[ \frac {x^{4} - 21 \, x^{2} - 20}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 0.86 \[ \ln \relax (x )-\frac {9 \ln \left (x^{2}+1\right )}{2}+4 \ln \left (x^{2}+2\right )+\frac {1}{2 x^{2}+2}-\frac {13}{2 \left (x^{2}+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 44, normalized size = 1.00 \[ -\frac {12 \, x^{2} + 11}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac {9}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
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 \[ 4\,\ln \left (x^2+2\right )-\frac {9\,\ln \left (x^2+1\right )}{2}+\ln \relax (x)-\frac {6\,x^2+\frac {11}{2}}{x^4+3\,x^2+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 41, normalized size = 0.93 \[ \frac {- 12 x^{2} - 11}{2 x^{4} + 6 x^{2} + 4} + \log {\relax (x )} - \frac {9 \log {\left (x^{2} + 1 \right )}}{2} + 4 \log {\left (x^{2} + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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