Optimal. Leaf size=69 \[ -\frac{x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}+\frac{11}{12 x^3}-\frac{1}{5 x^5}-\frac{23}{4 x}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
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Rubi [A] time = 0.0904842, antiderivative size = 69, 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} \[ -\frac{x \left (3-5 x^2\right )}{16 \left (x^4+3 x^2+2\right )}+\frac{11}{12 x^3}-\frac{1}{5 x^5}-\frac{23}{4 x}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}} \]
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 )^2} \, dx &=-\frac{x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-8+10 x^2-17 x^4+\frac{39 x^6}{4}-\frac{5 x^8}{4}}{x^6 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac{x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (-\frac{4}{x^6}+\frac{11}{x^4}-\frac{23}{x^2}+\frac{46}{1+x^2}-\frac{97}{4 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{5 x^5}+\frac{11}{12 x^3}-\frac{23}{4 x}-\frac{x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}+\frac{97}{16} \int \frac{1}{2+x^2} \, dx-\frac{23}{2} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{5 x^5}+\frac{11}{12 x^3}-\frac{23}{4 x}-\frac{x \left (3-5 x^2\right )}{16 \left (2+3 x^2+x^4\right )}-\frac{23}{2} \tan ^{-1}(x)+\frac{97 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0595521, size = 61, normalized size = 0.88 \[ \frac{1}{480} \left (\frac{30 x \left (5 x^2-3\right )}{x^4+3 x^2+2}+\frac{440}{x^3}-\frac{96}{x^5}-\frac{2760}{x}-5520 \tan ^{-1}(x)+1455 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 53, normalized size = 0.8 \begin{align*}{\frac{13\,x}{16\,{x}^{2}+32}}+{\frac{97\,\sqrt{2}}{32}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{23\,\arctan \left ( x \right ) }{2}}-{\frac{1}{5\,{x}^{5}}}+{\frac{11}{12\,{x}^{3}}}-{\frac{23}{4\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46622, size = 77, normalized size = 1.12 \begin{align*} \frac{97}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1305 \, x^{8} + 3965 \, x^{6} + 2148 \, x^{4} - 296 \, x^{2} + 96}{240 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} - \frac{23}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53035, size = 239, normalized size = 3.46 \begin{align*} -\frac{2610 \, x^{8} + 7930 \, x^{6} + 4296 \, x^{4} - 1455 \, \sqrt{2}{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 592 \, x^{2} + 5520 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )} \arctan \left (x\right ) + 192}{480 \,{\left (x^{9} + 3 \, x^{7} + 2 \, x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.24267, size = 61, normalized size = 0.88 \begin{align*} - \frac{23 \operatorname{atan}{\left (x \right )}}{2} + \frac{97 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{32} - \frac{1305 x^{8} + 3965 x^{6} + 2148 x^{4} - 296 x^{2} + 96}{240 x^{9} + 720 x^{7} + 480 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14244, size = 77, normalized size = 1.12 \begin{align*} \frac{97}{32} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{5 \, x^{3} - 3 \, x}{16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{345 \, x^{4} - 55 \, x^{2} + 12}{60 \, x^{5}} - \frac{23}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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