Optimal. Leaf size=64 \[ -\frac {1}{4 x^4}+\frac {11}{8 x^2}-\frac {11}{2} \log \left (x^2+1\right )+\frac {21}{8} \log \left (x^2+2\right )-\frac {9 x^2+5}{8 \left (x^4+3 x^2+2\right )}+\frac {23 \log (x)}{4} \]
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Rubi [A] time = 0.11, antiderivative size = 64, 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, 1628} \[ -\frac {9 x^2+5}{8 \left (x^4+3 x^2+2\right )}+\frac {11}{8 x^2}-\frac {1}{4 x^4}-\frac {11}{2} \log \left (x^2+1\right )+\frac {21}{8} \log \left (x^2+2\right )+\frac {23 \log (x)}{4} \]
Antiderivative was successfully verified.
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Rule 1628
Rule 1646
Rule 1663
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^5 \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^3 \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {5+9 x^2}{8 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-2+\frac {5 x}{2}-\frac {17 x^2}{4}+\frac {9 x^3}{4}}{x^3 \left (2+3 x+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {5+9 x^2}{8 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x^3}+\frac {11}{4 x^2}-\frac {23}{4 x}+\frac {11}{1+x}-\frac {21}{4 (2+x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 x^4}+\frac {11}{8 x^2}-\frac {5+9 x^2}{8 \left (2+3 x^2+x^4\right )}+\frac {23 \log (x)}{4}-\frac {11}{2} \log \left (1+x^2\right )+\frac {21}{8} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.88 \[ \frac {1}{8} \left (-\frac {2}{x^4}+\frac {11}{x^2}-44 \log \left (x^2+1\right )+21 \log \left (x^2+2\right )-\frac {9 x^2+5}{x^4+3 x^2+2}+46 \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 97, normalized size = 1.52 \[ \frac {2 \, x^{6} + 26 \, x^{4} + 16 \, x^{2} + 21 \, {\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \left (x^{2} + 2\right ) - 44 \, {\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \left (x^{2} + 1\right ) + 46 \, {\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )} \log \relax (x) - 4}{8 \, {\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 66, normalized size = 1.03 \[ \frac {23 \, x^{4} + 51 \, x^{2} + 36}{16 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {69 \, x^{4} - 22 \, x^{2} + 4}{16 \, x^{4}} + \frac {21}{8} \, \log \left (x^{2} + 2\right ) - \frac {11}{2} \, \log \left (x^{2} + 1\right ) + \frac {23}{8} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 50, normalized size = 0.78 \[ \frac {23 \ln \relax (x )}{4}-\frac {11 \ln \left (x^{2}+1\right )}{2}+\frac {21 \ln \left (x^{2}+2\right )}{8}+\frac {11}{8 x^{2}}-\frac {1}{4 x^{4}}+\frac {1}{2 x^{2}+2}-\frac {13}{8 \left (x^{2}+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 56, normalized size = 0.88 \[ \frac {x^{6} + 13 \, x^{4} + 8 \, x^{2} - 2}{4 \, {\left (x^{8} + 3 \, x^{6} + 2 \, x^{4}\right )}} + \frac {21}{8} \, \log \left (x^{2} + 2\right ) - \frac {11}{2} \, \log \left (x^{2} + 1\right ) + \frac {23}{8} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 55, normalized size = 0.86 \[ \frac {21\,\ln \left (x^2+2\right )}{8}-\frac {11\,\ln \left (x^2+1\right )}{2}+\frac {23\,\ln \relax (x)}{4}+\frac {\frac {x^6}{4}+\frac {13\,x^4}{4}+2\,x^2-\frac {1}{2}}{x^8+3\,x^6+2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 56, normalized size = 0.88 \[ \frac {23 \log {\relax (x )}}{4} - \frac {11 \log {\left (x^{2} + 1 \right )}}{2} + \frac {21 \log {\left (x^{2} + 2 \right )}}{8} + \frac {x^{6} + 13 x^{4} + 8 x^{2} - 2}{4 x^{8} + 12 x^{6} + 8 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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