Optimal. Leaf size=253 \[ -\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304} \]
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Rubi [A] time = 0.342795, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1669, 1664, 1169, 634, 618, 204, 628} \[ -\frac{25 x \left (x^2+5\right )}{144 \left (x^4+2 x^2+3\right )^2}-\frac{x \left (242 x^2+325\right )}{1728 \left (x^4+2 x^2+3\right )}-\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (55161 \sqrt{3}-59711\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{4608}-\frac{4}{27 x}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304} \]
Antiderivative was successfully verified.
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Rule 1669
Rule 1664
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left (3+2 x^2+x^4\right )^3} \, dx &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{128+30 x^2-\frac{250 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2048-\frac{56 x^2}{3}-\frac{1936 x^4}{3}}{x^2 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\int \left (\frac{2048}{3 x^2}-\frac{8 \left (173+166 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{1}{576} \int \frac{173+166 x^2}{3+2 x^2+x^4} \, dx\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\int \frac{173 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (173-166 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1152 \sqrt{6 \left (-1+\sqrt{3}\right )}}-\frac{\int \frac{173 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (173-166 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{1152 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{4608}-\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}-\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{2304}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )}{1152}+\frac{\sqrt{\frac{1}{3} \left (112597+57436 \sqrt{3}\right )} \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )}{1152}\\ &=-\frac{4}{27 x}-\frac{25 x \left (5+x^2\right )}{144 \left (3+2 x^2+x^4\right )^2}-\frac{x \left (325+242 x^2\right )}{1728 \left (3+2 x^2+x^4\right )}+\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (59711+55161 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{2304}-\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}+\frac{\sqrt{\frac{1}{3} \left (-59711+55161 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{4608}\\ \end{align*}
Mathematica [C] time = 0.378739, size = 140, normalized size = 0.55 \[ \frac{-\frac{12 \left (166 x^8+611 x^6+1412 x^4+1849 x^2+768\right )}{x \left (x^4+2 x^2+3\right )^2}+\frac{3 i \left (7 \sqrt{2}+332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}-\frac{3 i \left (7 \sqrt{2}-332 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{6912} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 424, normalized size = 1.7 \begin{align*} -{\frac{1}{27\, \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{121\,{x}^{7}}{32}}+{\frac{809\,{x}^{5}}{64}}+{\frac{419\,{x}^{3}}{16}}+{\frac{2475\,x}{64}} \right ) }-{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}+{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{325\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{27648}}-{\frac{7\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{9216}}-{\frac{ \left ( -650+650\,\sqrt{3} \right ) \sqrt{3}}{13824\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-14+14\,\sqrt{3}}{4608\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{173\,\sqrt{3}}{1728\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{27\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{166 \, x^{8} + 611 \, x^{6} + 1412 \, x^{4} + 1849 \, x^{2} + 768}{576 \,{\left (x^{9} + 4 \, x^{7} + 10 \, x^{5} + 12 \, x^{3} + 9 \, x\right )}} - \frac{1}{576} \, \int \frac{166 \, x^{2} + 173}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73342, size = 2529, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.607374, size = 73, normalized size = 0.29 \begin{align*} - \frac{166 x^{8} + 611 x^{6} + 1412 x^{4} + 1849 x^{2} + 768}{576 x^{9} + 2304 x^{7} + 5760 x^{5} + 6912 x^{3} + 5184 x} + \operatorname{RootSum}{\left (4174708211712 t^{4} + 15652880384 t^{2} + 37564641, \left ( t \mapsto t \log{\left (- \frac{98146713600 t^{3}}{11971753} - \frac{9639364864 t}{323237331} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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