Optimal. Leaf size=224 \[ \frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1678, 1169, 634, 618, 204, 628} \[ \frac {25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}+\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{96} \sqrt {\frac {1}{6} \left (11567+12897 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (12897 \sqrt {3}-11567\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1678
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {14+190 x^2}{3+2 x^2+x^4} \, dx\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {14 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (14-190 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {14 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (14-190 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {\left (7-95 \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}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {1}{288} \left (285+7 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{288} \left (285+7 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {\left (-7+95 \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}{96 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{144} \left (285+7 \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 )-\frac {1}{144} \left (285+7 \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 )\\ &=\frac {25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}-\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{48} \sqrt {\frac {1}{6} \left (-11567+12897 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{96} \sqrt {\frac {11567}{6}+\frac {4299 \sqrt {3}}{2}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.26, size = 115, normalized size = 0.51 \[ \frac {1}{48} \left (-\frac {50 x \left (x^2-1\right )}{x^4+2 x^2+3}+\frac {\left (95+44 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {\left (95-44 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 454, normalized size = 2.03 \[ -\frac {54052 \cdot 6160467^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \arctan \left (\frac {1}{29015889224422097862} \, \sqrt {19364129} 6160467^{\frac {3}{4}} \sqrt {174277161 \, x^{2} + 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}} {\left (95 \, \sqrt {3} \sqrt {2} - 7 \, \sqrt {2}\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{499478343426} \cdot 6160467^{\frac {3}{4}} {\left (95 \, \sqrt {3} \sqrt {2} x - 7 \, \sqrt {2} x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 54052 \cdot 6160467^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \arctan \left (\frac {1}{29015889224422097862} \, \sqrt {19364129} 6160467^{\frac {3}{4}} \sqrt {174277161 \, x^{2} - 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}} {\left (95 \, \sqrt {3} \sqrt {2} - 7 \, \sqrt {2}\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{499478343426} \cdot 6160467^{\frac {3}{4}} {\left (95 \, \sqrt {3} \sqrt {2} x - 7 \, \sqrt {2} x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 34855432200 \, x^{3} - 6160467^{\frac {1}{4}} {\left (11567 \, x^{4} + 23134 \, x^{2} + 12897 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 34701\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \log \left (174277161 \, x^{2} + 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}\right ) + 6160467^{\frac {1}{4}} {\left (11567 \, x^{4} + 23134 \, x^{2} + 12897 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 34701\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} \log \left (174277161 \, x^{2} - 6160467^{\frac {1}{4}} {\left (7 \, \sqrt {3} x - 285 \, x\right )} \sqrt {-149179599 \, \sqrt {3} + 498997827} + 174277161 \, \sqrt {3}\right ) - 34855432200 \, x}{33461214912 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.82, size = 565, normalized size = 2.52 \[ -\frac {1}{62208} \, \sqrt {2} {\left (95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1710 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 95 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 252 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{62208} \, \sqrt {2} {\left (95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1710 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 95 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 252 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) - \frac {1}{124416} \, \sqrt {2} {\left (1710 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 95 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 252 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + \frac {1}{124416} \, \sqrt {2} {\left (1710 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 95 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 95 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 1710 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 252 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 252 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {25 \, {\left (x^{3} - x\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 408, normalized size = 1.82 \[ \frac {139 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{288 \sqrt {2+2 \sqrt {3}}}+\frac {11 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{24 \sqrt {2+2 \sqrt {3}}}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{72 \sqrt {2+2 \sqrt {3}}}+\frac {139 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{288 \sqrt {2+2 \sqrt {3}}}+\frac {11 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{24 \sqrt {2+2 \sqrt {3}}}+\frac {7 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{72 \sqrt {2+2 \sqrt {3}}}+\frac {139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{576}+\frac {11 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{48}-\frac {139 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{576}-\frac {11 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{48}+\frac {-\frac {25}{24} x^{3}+\frac {25}{24} x}{x^{4}+2 x^{2}+3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {25 \, {\left (x^{3} - x\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {1}{24} \, \int \frac {95 \, x^{2} + 7}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 153, normalized size = 0.68 \[ \frac {\frac {25\,x}{24}-\frac {25\,x^3}{24}}{x^4+2\,x^2+3}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}\,13513{}\mathrm {i}}{15552\,\left (-\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}+\frac {13513\,\sqrt {2}\,x\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}}{31104\,\left (-\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {34701-\sqrt {2}\,40539{}\mathrm {i}}\,1{}\mathrm {i}}{144}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}\,13513{}\mathrm {i}}{15552\,\left (\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}-\frac {13513\,\sqrt {2}\,x\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}}{31104\,\left (\frac {1878307}{5184}+\frac {\sqrt {2}\,94591{}\mathrm {i}}{10368}\right )}\right )\,\sqrt {34701+\sqrt {2}\,40539{}\mathrm {i}}\,1{}\mathrm {i}}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.29, size = 1185, normalized size = 5.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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