Optimal. Leaf size=76 \[ \frac{x \left (3 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{23}{12 x^3}+\frac{11}{20 x^5}-\frac{1}{7 x^7}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.0999244, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \[ \frac{x \left (3 x^2+19\right )}{32 \left (x^4+3 x^2+2\right )}-\frac{23}{12 x^3}+\frac{11}{20 x^5}-\frac{1}{7 x^7}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]
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^8 \left (2+3 x^2+x^4\right )^2} \, dx &=\frac{x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-8+10 x^2-17 x^4+\frac{21 x^6}{2}-\frac{39 x^8}{8}-\frac{3 x^{10}}{8}}{x^8 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac{x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (-\frac{4}{x^8}+\frac{11}{x^6}-\frac{23}{x^4}+\frac{137}{4 x^2}-\frac{50}{1+x^2}+\frac{123}{8 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{137}{16 x}+\frac{x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}-\frac{123}{32} \int \frac{1}{2+x^2} \, dx+\frac{25}{2} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{7 x^7}+\frac{11}{20 x^5}-\frac{23}{12 x^3}+\frac{137}{16 x}+\frac{x \left (19+3 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.057174, size = 77, normalized size = 1.01 \[ \frac{3 x^3+19 x}{32 \left (x^4+3 x^2+2\right )}-\frac{23}{12 x^3}+\frac{11}{20 x^5}-\frac{1}{7 x^7}+\frac{137}{16 x}+\frac{25}{2} \tan ^{-1}(x)-\frac{123 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 58, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{32\,{x}^{2}+64}}-{\frac{123\,\sqrt{2}}{64}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{x}{2\,{x}^{2}+2}}+{\frac{25\,\arctan \left ( x \right ) }{2}}-{\frac{1}{7\,{x}^{7}}}+{\frac{11}{20\,{x}^{5}}}-{\frac{23}{12\,{x}^{3}}}+{\frac{137}{16\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53116, size = 84, normalized size = 1.11 \begin{align*} -\frac{123}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{29085 \, x^{10} + 81865 \, x^{8} + 40068 \, x^{6} - 7816 \, x^{4} + 2256 \, x^{2} - 960}{3360 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )}} + \frac{25}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56454, size = 271, normalized size = 3.57 \begin{align*} \frac{58170 \, x^{10} + 163730 \, x^{8} + 80136 \, x^{6} - 15632 \, x^{4} - 12915 \, \sqrt{2}{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 4512 \, x^{2} + 84000 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )} \arctan \left (x\right ) - 1920}{6720 \,{\left (x^{11} + 3 \, x^{9} + 2 \, x^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.263308, size = 66, normalized size = 0.87 \begin{align*} \frac{25 \operatorname{atan}{\left (x \right )}}{2} - \frac{123 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{64} + \frac{29085 x^{10} + 81865 x^{8} + 40068 x^{6} - 7816 x^{4} + 2256 x^{2} - 960}{3360 x^{11} + 10080 x^{9} + 6720 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07734, size = 84, normalized size = 1.11 \begin{align*} -\frac{123}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{3 \, x^{3} + 19 \, x}{32 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{14385 \, x^{6} - 3220 \, x^{4} + 924 \, x^{2} - 240}{1680 \, x^{7}} + \frac{25}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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