Optimal. Leaf size=69 \[ -\frac {1}{5 x^5}+\frac {11}{12 x^3}-\frac {x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}-\frac {23}{4 x}-\frac {23}{2} \tan ^{-1}(x)+\frac {97 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}} \]
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Rubi [A] time = 0.09, antiderivative size = 69, 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} \[ -\frac {x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}+\frac {11}{12 x^3}-\frac {1}{5 x^5}-\frac {23}{4 x}-\frac {23}{2} \tan ^{-1}(x)+\frac {97 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}} \]
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^6 \left (2+3 x^2+x^4\right )^2} \, dx &=-\frac {x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {-8+10 x^2-17 x^4+\frac {39 x^6}{4}-\frac {5 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac {x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-\frac {4}{x^6}+\frac {11}{x^4}-\frac {23}{x^2}+\frac {46}{1+x^2}-\frac {97}{4 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac {1}{5 x^5}+\frac {11}{12 x^3}-\frac {23}{4 x}-\frac {x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}+\frac {97}{16} \int \frac {1}{2+x^2} \, dx-\frac {23}{2} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{5 x^5}+\frac {11}{12 x^3}-\frac {23}{4 x}-\frac {x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac {23}{2} \tan ^{-1}(x)+\frac {97 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{16 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 61, normalized size = 0.88 \[ \frac {1}{480} \left (-\frac {96}{x^5}+\frac {440}{x^3}+\frac {30 x \left (5 x^2-3\right )}{x^4+3 x^2+2}-\frac {2760}{x}-5520 \tan ^{-1}(x)+1455 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 84, normalized size = 1.22 \[ -\frac {2610 \, x^{8} + 7930 \, x^{6} + 4296 \, x^{4} - 1455 \, \sqrt {2} {\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 592 \, x^{2} + 5520 \, {\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \relax (x) + 192}{480 \, {\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 57, normalized size = 0.83 \[ \frac {97}{32} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {5 \, x^{3} - 3 \, x}{16 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {345 \, x^{4} - 55 \, x^{2} + 12}{60 \, x^{5}} - \frac {23}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 53, normalized size = 0.77 \[ -\frac {x}{2 \left (x^{2}+1\right )}+\frac {13 x}{16 \left (x^{2}+2\right )}-\frac {23 \arctan \relax (x )}{2}+\frac {97 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{32}-\frac {23}{4 x}+\frac {11}{12 x^{3}}-\frac {1}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.76, size = 57, normalized size = 0.83 \[ \frac {97}{32} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1305 \, x^{8} + 3965 \, x^{6} + 2148 \, x^{4} - 296 \, x^{2} + 96}{240 \, {\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} - \frac {23}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 57, normalized size = 0.83 \[ \frac {97\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{32}-\frac {23\,\mathrm {atan}\relax (x)}{2}-\frac {\frac {87\,x^8}{16}+\frac {793\,x^6}{48}+\frac {179\,x^4}{20}-\frac {37\,x^2}{30}+\frac {2}{5}}{x^9+3\,x^7+2\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 61, normalized size = 0.88 \[ - \frac {23 \operatorname {atan}{\relax (x )}}{2} + \frac {97 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{32} + \frac {- 1305 x^{8} - 3965 x^{6} - 2148 x^{4} + 296 x^{2} - 96}{240 x^{9} + 720 x^{7} + 480 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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