Optimal. Leaf size=87 \[ \frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{54 x^2}+\frac{13}{108 x^4}-\frac{2}{27 x^6}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243} \]
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Rubi [A] time = 0.148986, antiderivative size = 87, 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} \[ \frac{25 \left (1-7 x^2\right )}{648 \left (x^4+2 x^2+3\right )}-\frac{13}{54 x^2}+\frac{13}{108 x^4}-\frac{2}{27 x^6}-\frac{61}{972} \log \left (x^4+2 x^2+3\right )-\frac{1237 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243} \]
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^7 \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^4 \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-7 x^2\right )}{648 \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{800 x^3}{81}-\frac{350 x^4}{81}}{x^4 \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (\frac{32}{9 x^4}-\frac{104}{27 x^3}+\frac{104}{27 x^2}+\frac{488}{243 x}-\frac{2 (1481+244 x)}{243 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{\operatorname{Subst}\left (\int \frac{1481+244 x}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{61}{972} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{1237 \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )}{1944}\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}+\frac{61 \log (x)}{243}-\frac{61}{972} \log \left (3+2 x^2+x^4\right )+\frac{1237}{972} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=-\frac{2}{27 x^6}+\frac{13}{108 x^4}-\frac{13}{54 x^2}+\frac{25 \left (1-7 x^2\right )}{648 \left (3+2 x^2+x^4\right )}-\frac{1237 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{1944 \sqrt{2}}+\frac{61 \log (x)}{243}-\frac{61}{972} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [C] time = 0.0689521, size = 110, normalized size = 1.26 \[ \frac{-\frac{300 \left (7 x^2-1\right )}{x^4+2 x^2+3}-\frac{1872}{x^2}+\frac{936}{x^4}-\frac{576}{x^6}+\sqrt{2} \left (-244 \sqrt{2}+1237 i\right ) \log \left (x^2-i \sqrt{2}+1\right )-\sqrt{2} \left (244 \sqrt{2}+1237 i\right ) \log \left (x^2+i \sqrt{2}+1\right )+1952 \log (x)}{7776} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 73, normalized size = 0.8 \begin{align*} -{\frac{1}{486\,{x}^{4}+972\,{x}^{2}+1458} \left ({\frac{525\,{x}^{2}}{4}}-{\frac{75}{4}} \right ) }-{\frac{61\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{972}}-{\frac{1237\,\sqrt{2}}{3888}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{2}{27\,{x}^{6}}}+{\frac{13}{108\,{x}^{4}}}-{\frac{13}{54\,{x}^{2}}}+{\frac{61\,\ln \left ( x \right ) }{243}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49926, size = 103, normalized size = 1.18 \begin{align*} -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{331 \, x^{8} + 209 \, x^{6} + 360 \, x^{4} - 138 \, x^{2} + 144}{648 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} - \frac{61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52743, size = 317, normalized size = 3.64 \begin{align*} -\frac{1986 \, x^{8} + 1254 \, x^{6} + 2160 \, x^{4} + 1237 \, \sqrt{2}{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 828 \, x^{2} + 244 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 976 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )} \log \left (x\right ) + 864}{3888 \,{\left (x^{10} + 2 \, x^{8} + 3 \, x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.236714, size = 85, normalized size = 0.98 \begin{align*} \frac{61 \log{\left (x \right )}}{243} - \frac{61 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{972} - \frac{1237 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{3888} - \frac{331 x^{8} + 209 x^{6} + 360 x^{4} - 138 x^{2} + 144}{648 x^{10} + 1296 x^{8} + 1944 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09403, size = 113, normalized size = 1.3 \begin{align*} -\frac{1237}{3888} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{122 \, x^{4} - 281 \, x^{2} + 441}{1944 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{671 \, x^{6} + 702 \, x^{4} - 351 \, x^{2} + 216}{2916 \, x^{6}} - \frac{61}{972} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{61}{486} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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