Optimal. Leaf size=72 \[ -\frac{x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (125 x^2+254\right )}{8 \left (x^4+3 x^2+2\right )}-\frac{369}{8} \tan ^{-1}(x)+\frac{267 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0678667, antiderivative size = 72, normalized size of antiderivative = 1., 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 = {1668, 1678, 1166, 203} \[ -\frac{x \left (51 x^2+50\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (125 x^2+254\right )}{8 \left (x^4+3 x^2+2\right )}-\frac{369}{8} \tan ^{-1}(x)+\frac{267 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-100+294 x^2+96 x^4-40 x^6}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{-816+660 x^2}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{369}{8} \int \frac{1}{1+x^2} \, dx+\frac{267}{4} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{x \left (50+51 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (254+125 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{369}{8} \tan ^{-1}(x)+\frac{267 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0605023, size = 55, normalized size = 0.76 \[ \frac{1}{8} \left (\frac{x \left (125 x^6+629 x^4+910 x^2+408\right )}{\left (x^4+3 x^2+2\right )^2}-369 \tan ^{-1}(x)+267 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 54, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{51\,{x}^{3}}{8}}+{\frac{77\,x}{4}} \right ) }+{\frac{267\,\sqrt{2}}{8}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{23\,{x}^{3}}{8}}-{\frac{25\,x}{8}} \right ) }-{\frac{369\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49012, size = 81, normalized size = 1.12 \begin{align*} \frac{267}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{369}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64032, size = 274, normalized size = 3.81 \begin{align*} \frac{125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 267 \, \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 ) - 369 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 408 \, 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 = 0.236653, size = 65, normalized size = 0.9 \begin{align*} \frac{125 x^{7} + 629 x^{5} + 910 x^{3} + 408 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac{369 \operatorname{atan}{\left (x \right )}}{8} + \frac{267 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10008, size = 68, normalized size = 0.94 \begin{align*} \frac{267}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{125 \, x^{7} + 629 \, x^{5} + 910 \, x^{3} + 408 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{369}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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