Optimal. Leaf size=75 \[ -\frac {\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac {\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac {413}{8} \tan ^{-1}(x)-\frac {191 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1668, 1678, 1676, 1166, 203} \begin {gather*} -\frac {\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac {\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac {413}{8} \tan ^{-1}(x)-\frac {191 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1166
Rule 1668
Rule 1676
Rule 1678
Rubi steps
\begin {align*} \int \frac {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {204-606 x^2-216 x^4+96 x^6-40 x^8}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (244+15 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {568-924 x^2+160 x^4}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (244+15 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \left (160+\frac {4 \left (62-351 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=5 x+\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (244+15 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{8} \int \frac {62-351 x^2}{2+3 x^2+x^4} \, dx\\ &=5 x+\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (244+15 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {413}{8} \int \frac {1}{1+x^2} \, dx-\frac {191}{2} \int \frac {1}{2+x^2} \, dx\\ &=5 x+\frac {x \left (102+103 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (244+15 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {413}{8} \tan ^{-1}(x)-\frac {191 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 60, normalized size = 0.80 \begin {gather*} \frac {1}{8} \left (\frac {x \left (40 x^8+225 x^6+231 x^4-76 x^2-124\right )}{\left (x^4+3 x^2+2\right )^2}+413 \tan ^{-1}(x)-382 \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 {x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.19, size = 104, normalized size = 1.39 \begin {gather*} \frac {40 \, x^{9} + 225 \, x^{7} + 231 \, x^{5} - 76 \, x^{3} - 382 \, \sqrt {2} {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 413 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \relax (x) - 124 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 53, normalized size = 0.71 \begin {gather*} -\frac {191}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x - \frac {15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac {413}{8} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.75 \begin {gather*} 5 x +\frac {413 \arctan \relax (x )}{8}-\frac {191 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{4}+\frac {-\frac {19}{8} x^{3}-\frac {21}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {16 \left (-\frac {1}{32} x^{3}+\frac {25}{16} x \right )}{\left (x^{2}+2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 63, normalized size = 0.84 \begin {gather*} -\frac {191}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x - \frac {15 \, x^{7} + 289 \, x^{5} + 556 \, x^{3} + 284 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac {413}{8} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 63, normalized size = 0.84 \begin {gather*} 5\,x+\frac {413\,\mathrm {atan}\relax (x)}{8}-\frac {191\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{4}-\frac {\frac {15\,x^7}{8}+\frac {289\,x^5}{8}+\frac {139\,x^3}{2}+\frac {71\,x}{2}}{x^8+6\,x^6+13\,x^4+12\,x^2+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 70, normalized size = 0.93 \begin {gather*} 5 x + \frac {- 15 x^{7} - 289 x^{5} - 556 x^{3} - 284 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac {413 \operatorname {atan}{\relax (x )}}{8} - \frac {191 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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