Optimal. Leaf size=66 \[ \frac {89 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac {1}{9} \log \left (x^4+2 x^2+3\right )+\frac {4 \log (x)}{9} \]
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Rubi [A] time = 0.11, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1663, 1646, 800, 634, 618, 204, 628} \[ \frac {25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac {1}{9} \log \left (x^4+2 x^2+3\right )+\frac {89 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {4 \log (x)}{9} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 800
Rule 1646
Rule 1663
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x \left (3+2 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 (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {\frac {32}{3}+\frac {70 x}{3}}{x \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \left (\frac {32}{9 x}-\frac {2 (-73+16 x)}{9 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{72} \operatorname {Subst}\left (\int \frac {-73+16 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )+\frac {89}{72} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{9} \log \left (3+2 x^2+x^4\right )-\frac {89}{36} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {89 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {4 \log (x)}{9}-\frac {1}{9} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 93, normalized size = 1.41 \[ \frac {1}{288} \left (-\sqrt {2} \left (16 \sqrt {2}+89 i\right ) \log \left (x^2-i \sqrt {2}+1\right )+\sqrt {2} \left (-16 \sqrt {2}+89 i\right ) \log \left (x^2+i \sqrt {2}+1\right )-\frac {300 \left (x^2-1\right )}{x^4+2 x^2+3}+128 \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 84, normalized size = 1.27 \[ \frac {89 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 150 \, x^{2} - 16 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \relax (x) + 150}{144 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 62, normalized size = 0.94 \[ \frac {89}{144} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {8 \, x^{4} - 59 \, x^{2} + 99}{72 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {2}{9} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.88 \[ \frac {89 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{144}+\frac {4 \ln \relax (x )}{9}-\frac {\ln \left (x^{4}+2 x^{2}+3\right )}{9}-\frac {\frac {75 x^{2}}{4}-\frac {75}{4}}{18 \left (x^{4}+2 x^{2}+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.68, size = 55, normalized size = 0.83 \[ \frac {89}{144} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {25 \, {\left (x^{2} - 1\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {2}{9} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 59, normalized size = 0.89 \[ \frac {4\,\ln \relax (x)}{9}-\frac {\ln \left (x^4+2\,x^2+3\right )}{9}-\frac {\frac {25\,x^2}{24}-\frac {25}{24}}{x^4+2\,x^2+3}+\frac {89\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 65, normalized size = 0.98 \[ \frac {25 - 25 x^{2}}{24 x^{4} + 48 x^{2} + 72} + \frac {4 \log {\relax (x )}}{9} - \frac {\log {\left (x^{4} + 2 x^{2} + 3 \right )}}{9} + \frac {89 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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