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 = 0.0913924, antiderivative size = 75, normalized size of antiderivative = 1., 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} \[ -\frac{\left (15 x^2+244\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (103 x^2+102\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+5 x+\frac{413}{8} \tan ^{-1}(x)-\frac{191 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1676
Rule 1166
Rule 203
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*}
Mathematica [A] time = 0.061187, size = 60, normalized size = 0.8 \[ \frac{1}{8} \left (\frac{x \left (40 x^8+225 x^6+231 x^4-76 x^2-124\right )}{\left (x^4+3 x^2+2\right )^2}+413 \tan ^{-1}(x)-382 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 56, normalized size = 0.8 \begin{align*} 5\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -1/32\,{x}^{3}+{\frac{25\,x}{16}} \right ) }-{\frac{191\,\sqrt{2}}{4}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{19\,{x}^{3}}{8}}-{\frac{21\,x}{8}} \right ) }+{\frac{413\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4955, size = 85, normalized size = 1.13 \begin{align*} -\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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62588, size = 285, normalized size = 3.8 \begin{align*} \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 \left (x\right ) - 124 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.240573, size = 68, normalized size = 0.91 \begin{align*} 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}{\left (x \right )}}{8} - \frac{191 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08504, size = 72, normalized size = 0.96 \begin{align*} -\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 \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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