Optimal. Leaf size=93 \[ -\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}+\frac{17}{24 x^3}-\frac{1}{10 x^5}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}} \]
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Rubi [A] time = 0.134206, antiderivative size = 93, 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} \[ -\frac{x \left (3-5 x^2\right )}{32 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (999 x^2+1771\right )}{128 \left (x^4+3 x^2+2\right )}+\frac{17}{24 x^3}-\frac{1}{10 x^5}-\frac{93}{16 x}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}} \]
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^6 \left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-16+20 x^2-34 x^4+\frac{81 x^6}{4}-\frac{25 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{32-88 x^2+184 x^4+\frac{681 x^6}{4}-\frac{999 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (\frac{16}{x^6}-\frac{68}{x^4}+\frac{186}{x^2}+\frac{116}{1+x^2}-\frac{2207}{4 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{93}{16 x}-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{29}{8} \int \frac{1}{1+x^2} \, dx-\frac{2207}{128} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{1}{10 x^5}+\frac{17}{24 x^3}-\frac{93}{16 x}-\frac{x \left (3-5 x^2\right )}{32 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (1771+999 x^2\right )}{128 \left (2+3 x^2+x^4\right )}+\frac{29}{8} \tan ^{-1}(x)-\frac{2207 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{128 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0788035, size = 73, normalized size = 0.78 \[ \frac{-\frac{2 \left (26145 x^{12}+137120 x^{10}+246477 x^8+170702 x^6+30816 x^4-3136 x^2+768\right )}{x^5 \left (x^4+3 x^2+2\right )^2}+13920 \tan ^{-1}(x)-33105 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{3840} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 68, normalized size = 0.7 \begin{align*} -{\frac{1}{16\, \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{311\,{x}^{3}}{8}}+{\frac{337\,x}{4}} \right ) }-{\frac{2207\,\sqrt{2}}{256}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{43\,{x}^{3}}{8}}-{\frac{45\,x}{8}} \right ) }+{\frac{29\,\arctan \left ( x \right ) }{8}}-{\frac{1}{10\,{x}^{5}}}+{\frac{17}{24\,{x}^{3}}}-{\frac{93}{16\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47447, size = 104, normalized size = 1.12 \begin{align*} -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{26145 \, x^{12} + 137120 \, x^{10} + 246477 \, x^{8} + 170702 \, x^{6} + 30816 \, x^{4} - 3136 \, x^{2} + 768}{1920 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} + \frac{29}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56934, size = 370, normalized size = 3.98 \begin{align*} -\frac{52290 \, x^{12} + 274240 \, x^{10} + 492954 \, x^{8} + 341404 \, x^{6} + 61632 \, x^{4} + 33105 \, \sqrt{2}{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 6272 \, x^{2} - 13920 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )} \arctan \left (x\right ) + 1536}{3840 \,{\left (x^{13} + 6 \, x^{11} + 13 \, x^{9} + 12 \, x^{7} + 4 \, x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.300799, size = 82, normalized size = 0.88 \begin{align*} \frac{29 \operatorname{atan}{\left (x \right )}}{8} - \frac{2207 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{256} - \frac{26145 x^{12} + 137120 x^{10} + 246477 x^{8} + 170702 x^{6} + 30816 x^{4} - 3136 x^{2} + 768}{1920 x^{13} + 11520 x^{11} + 24960 x^{9} + 23040 x^{7} + 7680 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12132, size = 90, normalized size = 0.97 \begin{align*} -\frac{2207}{256} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{999 \, x^{7} + 4768 \, x^{5} + 7291 \, x^{3} + 3554 \, x}{128 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{1395 \, x^{4} - 170 \, x^{2} + 24}{240 \, x^{5}} + \frac{29}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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