Optimal. Leaf size=72 \[ -\frac{x \left (12 x^2+11\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (217 x^2+335\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
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Rubi [A] time = 0.037454, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1678, 1178, 1166, 203} \[ -\frac{x \left (12 x^2+11\right )}{4 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (217 x^2+335\right )}{16 \left (x^4+3 x^2+2\right )}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1678
Rule 1178
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{\left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-38+80 x^2}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{-594+434 x^2}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac{257}{8} \int \frac{1}{1+x^2} \, dx+\frac{731}{16} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{x \left (11+12 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (335+217 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac{257}{8} \tan ^{-1}(x)+\frac{731 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0595557, size = 56, normalized size = 0.78 \[ \frac{1}{32} \left (\frac{2 x \left (217 x^6+986 x^4+1391 x^2+626\right )}{\left (x^4+3 x^2+2\right )^2}-1028 \tan ^{-1}(x)+731 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 53, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{155\,{x}^{3}}{16}}+{\frac{181\,x}{8}} \right ) }+{\frac{731\,\sqrt{2}}{32}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{31\,{x}^{3}}{8}}-{\frac{33\,x}{8}} \right ) }-{\frac{257\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51475, size = 81, normalized size = 1.12 \begin{align*} \frac{731}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{257}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57343, size = 281, normalized size = 3.9 \begin{align*} \frac{434 \, x^{7} + 1972 \, x^{5} + 2782 \, x^{3} + 731 \, \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 ) - 1028 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 1252 \, x}{32 \,{\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 = 0.231106, size = 65, normalized size = 0.9 \begin{align*} \frac{217 x^{7} + 986 x^{5} + 1391 x^{3} + 626 x}{16 x^{8} + 96 x^{6} + 208 x^{4} + 192 x^{2} + 64} - \frac{257 \operatorname{atan}{\left (x \right )}}{8} + \frac{731 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1064, size = 68, normalized size = 0.94 \begin{align*} \frac{731}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{217 \, x^{7} + 986 \, x^{5} + 1391 \, x^{3} + 626 \, x}{16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{257}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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