Optimal. Leaf size=62 \[ -\frac {1}{3 x^3}-\frac {x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}+\frac {11}{4 x}+\frac {21}{2} \tan ^{-1}(x)-\frac {71 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \begin {gather*} -\frac {x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}-\frac {1}{3 x^3}+\frac {11}{4 x}+\frac {21}{2} \tan ^{-1}(x)-\frac {71 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1664
Rule 1669
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^4 \left (2+3 x^2+x^4\right )^2} \, dx &=-\frac {x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {-8+10 x^2-\frac {39 x^4}{2}+\frac {9 x^6}{2}}{x^4 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac {x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-\frac {4}{x^4}+\frac {11}{x^2}-\frac {42}{1+x^2}+\frac {71}{2 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac {1}{3 x^3}+\frac {11}{4 x}-\frac {x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac {71}{8} \int \frac {1}{2+x^2} \, dx+\frac {21}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{3 x^3}+\frac {11}{4 x}-\frac {x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {21}{2} \tan ^{-1}(x)-\frac {71 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 0.90 \begin {gather*} \frac {1}{48} \left (-\frac {16}{x^3}-\frac {6 x \left (9 x^2+5\right )}{x^4+3 x^2+2}+\frac {132}{x}+504 \tan ^{-1}(x)-213 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\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 {4+x^2+3 x^4+5 x^6}{x^4 \left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.61, size = 79, normalized size = 1.27 \begin {gather*} \frac {78 \, x^{6} + 350 \, x^{4} - 213 \, \sqrt {2} {\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 216 \, x^{2} + 504 \, {\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \relax (x) - 32}{48 \, {\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 52, normalized size = 0.84 \begin {gather*} -\frac {71}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {9 \, x^{3} + 5 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac {33 \, x^{2} - 4}{12 \, x^{3}} + \frac {21}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 48, normalized size = 0.77 \begin {gather*} \frac {x}{2 x^{2}+2}-\frac {13 x}{8 \left (x^{2}+2\right )}+\frac {21 \arctan \relax (x )}{2}-\frac {71 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{16}+\frac {11}{4 x}-\frac {1}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 52, normalized size = 0.84 \begin {gather*} -\frac {71}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {39 \, x^{6} + 175 \, x^{4} + 108 \, x^{2} - 16}{24 \, {\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} + \frac {21}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 51, normalized size = 0.82 \begin {gather*} \frac {21\,\mathrm {atan}\relax (x)}{2}-\frac {71\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{16}+\frac {\frac {13\,x^6}{8}+\frac {175\,x^4}{24}+\frac {9\,x^2}{2}-\frac {2}{3}}{x^7+3\,x^5+2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 56, normalized size = 0.90 \begin {gather*} \frac {21 \operatorname {atan}{\relax (x )}}{2} - \frac {71 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{16} + \frac {39 x^{6} + 175 x^{4} + 108 x^{2} - 16}{24 x^{7} + 72 x^{5} + 48 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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