Optimal. Leaf size=80 \[ \frac{25 \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}+\frac{13}{54 x^2}-\frac{1}{9 x^4}-\frac{13}{108} \log \left (x^4+2 x^2+3\right )+\frac{125 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}+\frac{13 \log (x)}{27} \]
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Rubi [A] time = 0.136748, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1663, 1646, 1628, 634, 618, 204, 628} \[ \frac{25 \left (5 x^2+7\right )}{216 \left (x^4+2 x^2+3\right )}+\frac{13}{54 x^2}-\frac{1}{9 x^4}-\frac{13}{108} \log \left (x^4+2 x^2+3\right )+\frac{125 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{216 \sqrt{2}}+\frac{13 \log (x)}{27} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1646
Rule 1628
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^5 \left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{4+x+3 x^2+5 x^3}{x^3 \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{\frac{32}{3}-\frac{40 x}{9}+\frac{200 x^2}{27}+\frac{250 x^3}{27}}{x^3 \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (\frac{32}{9 x^3}-\frac{104}{27 x^2}+\frac{104}{27 x}-\frac{2 (-73+52 x)}{27 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{13 \log (x)}{27}-\frac{1}{216} \operatorname{Subst}\left (\int \frac{-73+52 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{13 \log (x)}{27}-\frac{13}{108} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )+\frac{125}{216} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{13 \log (x)}{27}-\frac{13}{108} \log \left (3+2 x^2+x^4\right )-\frac{125}{108} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=-\frac{1}{9 x^4}+\frac{13}{54 x^2}+\frac{25 \left (7+5 x^2\right )}{216 \left (3+2 x^2+x^4\right )}+\frac{125 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{216 \sqrt{2}}+\frac{13 \log (x)}{27}-\frac{13}{108} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [C] time = 0.0613122, size = 105, normalized size = 1.31 \[ \frac{1}{864} \left (\frac{100 \left (5 x^2+7\right )}{x^4+2 x^2+3}+\frac{208}{x^2}-\frac{96}{x^4}-\sqrt{2} \left (52 \sqrt{2}+125 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (-52 \sqrt{2}+125 i\right ) \log \left (x^2+i \sqrt{2}+1\right )+416 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 68, normalized size = 0.9 \begin{align*} -{\frac{1}{54\,{x}^{4}+108\,{x}^{2}+162} \left ( -{\frac{125\,{x}^{2}}{4}}-{\frac{175}{4}} \right ) }-{\frac{13\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{108}}+{\frac{125\,\sqrt{2}}{432}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{1}{9\,{x}^{4}}}+{\frac{13}{54\,{x}^{2}}}+{\frac{13\,\ln \left ( x \right ) }{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4745, size = 96, normalized size = 1.2 \begin{align*} \frac{125}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{59 \, x^{6} + 85 \, x^{4} + 36 \, x^{2} - 24}{72 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )}} - \frac{13}{108} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{13}{54} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49019, size = 289, normalized size = 3.61 \begin{align*} \frac{354 \, x^{6} + 510 \, x^{4} + 125 \, \sqrt{2}{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 216 \, x^{2} - 52 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 208 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )} \log \left (x\right ) - 144}{432 \,{\left (x^{8} + 2 \, x^{6} + 3 \, x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.214274, size = 80, normalized size = 1. \begin{align*} \frac{13 \log{\left (x \right )}}{27} - \frac{13 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{108} + \frac{125 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{432} + \frac{59 x^{6} + 85 x^{4} + 36 x^{2} - 24}{72 x^{8} + 144 x^{6} + 216 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08324, size = 107, normalized size = 1.34 \begin{align*} \frac{125}{432} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{26 \, x^{4} + 177 \, x^{2} + 253}{216 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{39 \, x^{4} - 26 \, x^{2} + 12}{108 \, x^{4}} - \frac{13}{108} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{13}{54} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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