Optimal. Leaf size=58 \[ -\frac {23 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac {5}{4} \log \left (x^4+2 x^2+3\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1663, 1660, 634, 618, 204, 628} \begin {gather*} \frac {25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac {5}{4} \log \left (x^4+2 x^2+3\right )-\frac {23 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x \left (4+x^2+3 x^4+5 x^6\right )}{\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}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-6+40 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {5}{4} \operatorname {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac {23}{8} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {5}{4} \log \left (3+2 x^2+x^4\right )+\frac {23}{4} \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 )}{8 \left (3+2 x^2+x^4\right )}-\frac {23 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {5}{4} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.00 \begin {gather*} -\frac {23 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac {5}{4} \log \left (x^4+2 x^2+3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.33, size = 70, normalized size = 1.21 \begin {gather*} -\frac {23 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 50 \, x^{2} - 20 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 50}{16 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 49, normalized size = 0.84 \begin {gather*} -\frac {23}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {25 \, {\left (x^{2} + 1\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {5}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.93 \begin {gather*} -\frac {23 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{16}+\frac {5 \ln \left (x^{4}+2 x^{2}+3\right )}{4}+\frac {\frac {25 x^{2}}{4}+\frac {25}{4}}{2 x^{4}+4 x^{2}+6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 49, normalized size = 0.84 \begin {gather*} -\frac {23}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {25 \, {\left (x^{2} + 1\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {5}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 69, normalized size = 1.19 \begin {gather*} \frac {5\,\ln \left (x^4+2\,x^2+3\right )}{4}+\frac {25\,x^2}{8\,\left (x^4+2\,x^2+3\right )}+\frac {25}{8\,\left (x^4+2\,x^2+3\right )}-\frac {23\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 60, normalized size = 1.03 \begin {gather*} \frac {25 x^{2} + 25}{8 x^{4} + 16 x^{2} + 24} + \frac {5 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac {23 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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