Optimal. Leaf size=86 \[ -\frac{x \left (9 x^2+5\right )}{16 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (571 x^2+951\right )}{64 \left (x^4+3 x^2+2\right )}-\frac{1}{6 x^3}+\frac{17}{8 x}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{64 \sqrt{2}} \]
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Rubi [A] time = 0.118885, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \[ -\frac{x \left (9 x^2+5\right )}{16 \left (x^4+3 x^2+2\right )^2}+\frac{x \left (571 x^2+951\right )}{64 \left (x^4+3 x^2+2\right )}-\frac{1}{6 x^3}+\frac{17}{8 x}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{64 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1669
Rule 1664
Rule 203
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^4 \left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (5+9 x^2\right )}{16 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-16+20 x^2-\frac{73 x^4}{2}+\frac{45 x^6}{2}}{x^4 \left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (5+9 x^2\right )}{16 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (951+571 x^2\right )}{64 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{32-88 x^2-\frac{573 x^4}{2}+\frac{571 x^6}{2}}{x^4 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac{x \left (5+9 x^2\right )}{16 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (951+571 x^2\right )}{64 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (\frac{16}{x^4}-\frac{68}{x^2}-\frac{452}{1+x^2}+\frac{1611}{2 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{6 x^3}+\frac{17}{8 x}-\frac{x \left (5+9 x^2\right )}{16 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (951+571 x^2\right )}{64 \left (2+3 x^2+x^4\right )}-\frac{113}{8} \int \frac{1}{1+x^2} \, dx+\frac{1611}{64} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{1}{6 x^3}+\frac{17}{8 x}-\frac{x \left (5+9 x^2\right )}{16 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (951+571 x^2\right )}{64 \left (2+3 x^2+x^4\right )}-\frac{113}{8} \tan ^{-1}(x)+\frac{1611 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{64 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0603144, size = 78, normalized size = 0.91 \[ \frac{1}{384} \left (-\frac{24 x \left (9 x^2+5\right )}{\left (x^4+3 x^2+2\right )^2}+\frac{6 x \left (571 x^2+951\right )}{x^4+3 x^2+2}-\frac{64}{x^3}+\frac{816}{x}-5424 \tan ^{-1}(x)+4833 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 64, normalized size = 0.7 \begin{align*}{\frac{1}{8\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{259\,{x}^{3}}{8}}+{\frac{285\,x}{4}} \right ) }+{\frac{1611\,\sqrt{2}}{128}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{39\,{x}^{3}}{8}}-{\frac{41\,x}{8}} \right ) }-{\frac{113\,\arctan \left ( x \right ) }{8}}-{\frac{1}{6\,{x}^{3}}}+{\frac{17}{8\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54805, size = 97, normalized size = 1.13 \begin{align*} \frac{1611}{128} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{2121 \, x^{10} + 10408 \, x^{8} + 16989 \, x^{6} + 10126 \, x^{4} + 1248 \, x^{2} - 128}{192 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )}} - \frac{113}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54364, size = 336, normalized size = 3.91 \begin{align*} \frac{4242 \, x^{10} + 20816 \, x^{8} + 33978 \, x^{6} + 20252 \, x^{4} + 4833 \, \sqrt{2}{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 2496 \, x^{2} - 5424 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )} \arctan \left (x\right ) - 256}{384 \,{\left (x^{11} + 6 \, x^{9} + 13 \, x^{7} + 12 \, x^{5} + 4 \, x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.277716, size = 76, normalized size = 0.88 \begin{align*} - \frac{113 \operatorname{atan}{\left (x \right )}}{8} + \frac{1611 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{128} + \frac{2121 x^{10} + 10408 x^{8} + 16989 x^{6} + 10126 x^{4} + 1248 x^{2} - 128}{192 x^{11} + 1152 x^{9} + 2496 x^{7} + 2304 x^{5} + 768 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11011, size = 84, normalized size = 0.98 \begin{align*} \frac{1611}{128} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{571 \, x^{7} + 2664 \, x^{5} + 3959 \, x^{3} + 1882 \, x}{64 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{51 \, x^{2} - 4}{24 \, x^{3}} - \frac{113}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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