Optimal. Leaf size=238 \[ \frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1668, 1678, 1169, 634, 618, 204, 628} \[ \frac {x \left (238-59 x^2\right )}{64 \left (x^4+2 x^2+3\right )}-\frac {25 x \left (x^2+3\right )}{16 \left (x^4+2 x^2+3\right )^2}+\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{512} \sqrt {3 \left (48835+32827 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{256} \sqrt {3 \left (32827 \sqrt {3}-48835\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1668
Rule 1678
Rubi steps
\begin {align*} \int \frac {x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {450-750 x^2-672 x^4+480 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {-9936+18792 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {-9936 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (-9936-18792 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {-9936 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (-9936-18792 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {1}{256} \left (261-46 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{256} \left (261-46 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{256} \left (\sqrt {\frac {3}{2 \left (-1+\sqrt {3}\right )}} \left (46+87 \sqrt {3}\right )\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-\frac {1}{256} \left (\sqrt {\frac {3}{2 \left (-1+\sqrt {3}\right )}} \left (46+87 \sqrt {3}\right )\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\\ &=-\frac {25 x \left (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{128} \left (-261+46 \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}{128} \left (-261+46 \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 (3+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {x \left (238-59 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac {1}{256} \sqrt {3 \left (-48835+32827 \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}{256} \sqrt {3 \left (-48835+32827 \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}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )-\frac {1}{512} \sqrt {146505+98481 \sqrt {3}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.30, size = 129, normalized size = 0.54 \[ \frac {1}{256} \left (\frac {4 x \left (-59 x^6+120 x^4+199 x^2+414\right )}{\left (x^4+2 x^2+3\right )^2}+\frac {3 \left (174+133 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {3 \left (174-133 i \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 546, normalized size = 2.29 \[ -\frac {1914264223824 \, x^{7} - 3893418760320 \, x^{5} + 164728 \cdot 29095522083^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \arctan \left (\frac {1}{1214880276996365518761363} \cdot 29095522083^{\frac {3}{4}} \sqrt {2027822271} \sqrt {2027822271 \, x^{2} + 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}} {\left (46 \, \sqrt {3} + 261\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{599105895211053} \cdot 29095522083^{\frac {3}{4}} {\left (46 \, \sqrt {3} x + 261 \, x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 164728 \cdot 29095522083^{\frac {1}{4}} \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \arctan \left (\frac {1}{1214880276996365518761363} \cdot 29095522083^{\frac {3}{4}} \sqrt {2027822271} \sqrt {2027822271 \, x^{2} - 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}} {\left (46 \, \sqrt {3} + 261\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} - \frac {1}{599105895211053} \cdot 29095522083^{\frac {3}{4}} {\left (46 \, \sqrt {3} x + 261 \, x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 6456586110864 \, x^{3} + 29095522083^{\frac {1}{4}} {\left (48835 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 98481 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \log \left (2027822271 \, x^{2} + 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}\right ) - 29095522083^{\frac {1}{4}} {\left (48835 \, \sqrt {3} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 98481 \, \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} \log \left (2027822271 \, x^{2} - 29095522083^{\frac {1}{4}} {\left (87 \, \sqrt {3} \sqrt {2} x + 46 \, \sqrt {2} x\right )} \sqrt {-1603106545 \, \sqrt {3} + 3232835787} + 2027822271 \, \sqrt {3}\right ) - 13432294723104 \, x}{2076490005504 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.60, size = 577, normalized size = 2.42 \[ -\frac {1}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{18432} \, \sqrt {2} {\left (29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 522 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 29 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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}{36864} \, \sqrt {2} {\left (522 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 29 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 29 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 522 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 552 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 552 \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 {59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 418, normalized size = 1.76 \[ \frac {307 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}+\frac {399 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {23 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}+\frac {307 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}+\frac {399 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{512 \sqrt {2+2 \sqrt {3}}}-\frac {23 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{32 \sqrt {2+2 \sqrt {3}}}+\frac {307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {399 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {307 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}-\frac {399 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{1024}+\frac {-\frac {59}{64} x^{7}+\frac {15}{8} x^{5}+\frac {199}{64} x^{3}+\frac {207}{32} x}{\left (x^{4}+2 x^{2}+3\right )^{2}} \]
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 \[ -\frac {59 \, x^{7} - 120 \, x^{5} - 199 \, x^{3} - 414 \, x}{64 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac {3}{64} \, \int \frac {87 \, x^{2} - 46}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 173, normalized size = 0.73 \[ \frac {-\frac {59\,x^7}{64}+\frac {15\,x^5}{8}+\frac {199\,x^3}{64}+\frac {207\,x}{32}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}+\frac {61773\,\sqrt {2}\,x\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010-\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,61773{}\mathrm {i}}{131072\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}-\frac {61773\,\sqrt {2}\,x\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}}{262144\,\left (-\frac {56892933}{131072}+\frac {\sqrt {2}\,4262337{}\mathrm {i}}{65536}\right )}\right )\,\sqrt {293010+\sqrt {2}\,123546{}\mathrm {i}}\,1{}\mathrm {i}}{256} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.31, size = 1198, normalized size = 5.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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