Optimal. Leaf size=72 \[ \frac {x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {317}{8} \tan ^{-1}(x)-\frac {447 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1682, 1692,
1180, 209} \begin {gather*} \frac {317 \text {ArcTan}(x)}{8}-\frac {447 \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {x \left (25 x^2+24\right )}{4 \left (x^4+3 x^2+2\right )^2}-\frac {x \left (130 x^2+211\right )}{8 \left (x^4+3 x^2+2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1180
Rule 1682
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac {x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {48-154 x^2-40 x^4}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {748-520 x^2}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {317}{8} \int \frac {1}{1+x^2} \, dx-\frac {447}{8} \int \frac {1}{2+x^2} \, dx\\ &=\frac {x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac {317}{8} \tan ^{-1}(x)-\frac {447 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.78 \begin {gather*} \frac {1}{16} \left (-\frac {2 x \left (374+843 x^2+601 x^4+130 x^6\right )}{\left (2+3 x^2+x^4\right )^2}+634 \tan ^{-1}(x)-447 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 53, normalized size = 0.74
method | result | size |
risch | \(\frac {-\frac {65}{4} x^{7}-\frac {601}{8} x^{5}-\frac {843}{8} x^{3}-\frac {187}{4} x}{\left (x^{4}+3 x^{2}+2\right )^{2}}+\frac {317 \arctan \left (x \right )}{8}-\frac {447 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{16}\) | \(50\) |
default | \(\frac {-\frac {27}{8} x^{3}-\frac {29}{8} x}{\left (x^{2}+1\right )^{2}}+\frac {317 \arctan \left (x \right )}{8}-\frac {\frac {103}{8} x^{3}+\frac {129}{4} x}{\left (x^{2}+2\right )^{2}}-\frac {447 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{16}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 60, normalized size = 0.83 \begin {gather*} -\frac {447}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac {317}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 99, normalized size = 1.38 \begin {gather*} -\frac {260 \, x^{7} + 1202 \, x^{5} + 1686 \, x^{3} + 447 \, \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 ) - 634 \, {\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 748 \, x}{16 \, {\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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Sympy [A]
time = 0.10, size = 66, normalized size = 0.92 \begin {gather*} \frac {- 130 x^{7} - 601 x^{5} - 843 x^{3} - 374 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac {317 \operatorname {atan}{\left (x \right )}}{8} - \frac {447 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.32, size = 50, normalized size = 0.69 \begin {gather*} -\frac {447}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac {317}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 60, normalized size = 0.83 \begin {gather*} \frac {317\,\mathrm {atan}\left (x\right )}{8}-\frac {447\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{16}-\frac {\frac {65\,x^7}{4}+\frac {601\,x^5}{8}+\frac {843\,x^3}{8}+\frac {187\,x}{4}}{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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