Optimal. Leaf size=262 \[ \frac{25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}-\frac{4}{81 x^3}+\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}-\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}+\frac{7}{27 x}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736} \]
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Rubi [A] time = 0.365888, antiderivative size = 262, 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} \[ \frac{25 x \left (5 x^2+7\right )}{432 \left (x^4+2 x^2+3\right )^2}+\frac{x \left (1025 x^2+1474\right )}{5184 \left (x^4+2 x^2+3\right )}-\frac{4}{81 x^3}+\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}-\frac{\sqrt{\frac{1}{3} \left (11240451 \sqrt{3}-10004741\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )}{41472}+\frac{7}{27 x}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736} \]
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^4 \left (3+2 x^2+x^4\right )^3} \, dx &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{128-\frac{160 x^2}{3}+50 x^4+\frac{1250 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2048-\frac{6656 x^2}{3}+\frac{2576 x^4}{9}+\frac{8200 x^6}{9}}{x^4 \left (3+2 x^2+x^4\right )} \, dx}{4608}\\ &=\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \left (\frac{2048}{3 x^4}-\frac{3584}{3 x^2}+\frac{8 \left (2242+2369 x^2\right )}{9 \left (3+2 x^2+x^4\right )}\right ) \, dx}{4608}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2242+2369 x^2}{3+2 x^2+x^4} \, dx}{5184}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{2242 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (2242-2369 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{10368 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{2242 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (2242-2369 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{10368 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\left (2242-2369 \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}{20736 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\left (7107+2242 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{62208}+\frac{\left (7107+2242 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{62208}+\frac{\left (-2242+2369 \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}{20736 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}+\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\left (7107+2242 \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 )}{31104}-\frac{\left (7107+2242 \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 )}{31104}\\ &=-\frac{4}{81 x^3}+\frac{7}{27 x}+\frac{25 x \left (7+5 x^2\right )}{432 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (1474+1025 x^2\right )}{5184 \left (3+2 x^2+x^4\right )}-\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{\frac{1}{3} \left (10004741+11240451 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )}{20736}+\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}-\frac{\sqrt{-\frac{10004741}{12}+\frac{3746817 \sqrt{3}}{4}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )}{20736}\\ \end{align*}
Mathematica [C] time = 0.331905, size = 139, normalized size = 0.53 \[ \frac{\frac{4 \left (2369 x^{10}+8644 x^8+19939 x^6+20090 x^4+9024 x^2-2304\right )}{x^3 \left (x^4+2 x^2+3\right )^2}+\frac{\left (4738+127 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{\left (4738-127 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}}{20736} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 429, normalized size = 1.6 \begin{align*}{\frac{1}{27\, \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{1025\,{x}^{7}}{192}}+{\frac{881\,{x}^{5}}{48}}+{\frac{7523\,{x}^{3}}{192}}+{\frac{1087\,x}{32}} \right ) }+{\frac{4865\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{248832}}+{\frac{127\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{82944}}+{\frac{ \left ( -9730+9730\,\sqrt{3} \right ) \sqrt{3}}{124416\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-254+254\,\sqrt{3}}{41472\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{1121\,\sqrt{3}}{7776\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4865\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{248832}}-{\frac{127\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{82944}}+{\frac{ \left ( -9730+9730\,\sqrt{3} \right ) \sqrt{3}}{124416\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-254+254\,\sqrt{3}}{41472\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{1121\,\sqrt{3}}{7776\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4}{81\,{x}^{3}}}+{\frac{7}{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{2369 \, x^{10} + 8644 \, x^{8} + 19939 \, x^{6} + 20090 \, x^{4} + 9024 \, x^{2} - 2304}{5184 \,{\left (x^{11} + 4 \, x^{9} + 10 \, x^{7} + 12 \, x^{5} + 9 \, x^{3}\right )}} + \frac{1}{5184} \, \int \frac{2369 \, x^{2} + 2242}{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 = 1.74976, size = 2952, normalized size = 11.27 \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 = 0.6269, size = 80, normalized size = 0.31 \begin{align*} \operatorname{RootSum}{\left (338151365148672 t^{4} + 2622682824704 t^{2} + 19257390441, \left ( t \mapsto t \log{\left (\frac{357010935644160 t^{3}}{182097141061} + \frac{26016957890816 t}{1638874269549} + x \right )} \right )\right )} + \frac{2369 x^{10} + 8644 x^{8} + 19939 x^{6} + 20090 x^{4} + 9024 x^{2} - 2304}{5184 x^{11} + 20736 x^{9} + 51840 x^{7} + 62208 x^{5} + 46656 x^{3}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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