Optimal. Leaf size=225 \[ 5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1682, 1690,
1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \sqrt {3}\right )} \text {ArcTan}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (12899 \sqrt {3}-19291\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {25 \left (x^2+1\right ) x}{8 \left (x^4+2 x^2+3\right )}+5 x \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1682
Rule 1690
Rubi steps
\begin {align*} \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {-150-186 x^2+240 x^4}{3+2 x^2+x^4} \, dx\\ &=\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (240-\frac {6 \left (145+111 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{8} \int \frac {145+111 x^2}{3+2 x^2+x^4} \, dx\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {\int \frac {145 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (145-111 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{16 \sqrt {6 \left (-1+\sqrt {3}\right )}}-\frac {\int \frac {145 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (145-111 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{16 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{96} \left (333+145 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{96} \left (333+145 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx-\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \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+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \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\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{48} \left (333+145 \sqrt {3}\right ) \text {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}{48} \left (333+145 \sqrt {3}\right ) \text {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 )\\ &=5 x+\frac {25 x \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \sqrt {\frac {1}{6} \left (19291+12899 \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}{16} \sqrt {\frac {1}{6} \left (19291+12899 \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}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (-19291+12899 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 121, normalized size = 0.54 \begin {gather*} 5 x+\frac {25 \left (x+x^3\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {\left (-34 i+111 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}-\frac {\left (34 i+111 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{16 \sqrt {2+2 i \sqrt {2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 281, normalized size = 1.25
method | result | size |
risch | \(5 x +\frac {\frac {25}{8} x^{3}+\frac {25}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (-111 \textit {\_R}^{2}-145\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{32}\) | \(64\) |
default | \(5 x -\frac {-\frac {25}{8} x^{3}-\frac {25}{8} x}{x^{4}+2 x^{2}+3}-\frac {\left (94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-51 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{192}-\frac {\left (290 \sqrt {3}+\frac {\left (94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-51 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+51 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{192}-\frac {\left (290 \sqrt {3}-\frac {\left (-94 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+51 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}\) | \(281\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (158) = 316\).
time = 0.37, size = 461, normalized size = 2.05 \begin {gather*} \frac {98680445760 \, x^{5} + 31876 \cdot 499152603^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \arctan \left (\frac {1}{2453286601800494203302} \cdot 499152603^{\frac {3}{4}} \sqrt {7969} \sqrt {11933241279921 \, x^{2} + 38697 \cdot 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 11933241279921 \, \sqrt {3}} {\left (111 \, \sqrt {3} \sqrt {2} - 145 \, \sqrt {2}\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{7955494186614} \cdot 499152603^{\frac {3}{4}} {\left (111 \, \sqrt {3} \sqrt {2} x - 145 \, \sqrt {2} x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) + 31876 \cdot 499152603^{\frac {1}{4}} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \arctan \left (\frac {1}{2453286601800494203302} \cdot 499152603^{\frac {3}{4}} \sqrt {7969} \sqrt {11933241279921 \, x^{2} - 38697 \cdot 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 11933241279921 \, \sqrt {3}} {\left (111 \, \sqrt {3} \sqrt {2} - 145 \, \sqrt {2}\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{7955494186614} \cdot 499152603^{\frac {3}{4}} {\left (111 \, \sqrt {3} \sqrt {2} x - 145 \, \sqrt {2} x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 259036170120 \, x^{3} + 499152603^{\frac {1}{4}} {\left (19291 \, x^{4} + 38582 \, x^{2} - 12899 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 57873\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \log \left (1497457809 \, x^{2} + \frac {38697}{7969} \cdot 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 1497457809 \, \sqrt {3}\right ) - 499152603^{\frac {1}{4}} {\left (19291 \, x^{4} + 38582 \, x^{2} - 12899 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 57873\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} \log \left (1497457809 \, x^{2} - \frac {38697}{7969} \cdot 499152603^{\frac {1}{4}} {\left (145 \, \sqrt {3} x - 333 \, x\right )} \sqrt {248834609 \, \sqrt {3} + 499152603} + 1497457809 \, \sqrt {3}\right ) + 357716615880 \, x}{19736089152 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 51, normalized size = 0.23 \begin {gather*} 5 x + \frac {25 x^{3} + 25 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (3145728 t^{4} + 39507968 t^{2} + 166384201, \left ( t \mapsto t \log {\left (- \frac {9240576 t^{3}}{102792131} - \frac {95003488 t}{102792131} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs.
\(2 (158) = 316\).
time = 5.36, size = 566, normalized size = 2.52 \begin {gather*} \frac {1}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \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}{6912} \, \sqrt {2} {\left (37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 666 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 37 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 1740 \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}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \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}{13824} \, \sqrt {2} {\left (666 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 37 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 37 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 666 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 1740 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 1740 \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 ) + 5 \, x + \frac {25 \, {\left (x^{3} + x\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 156, normalized size = 0.69 \begin {gather*} 5\,x+\frac {\frac {25\,x^3}{8}+\frac {25\,x}{8}}{x^4+2\,x^2+3}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}+\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (-\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873-\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,7969{}\mathrm {i}}{576\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}-\frac {7969\,\sqrt {2}\,x\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}}{1152\,\left (\frac {374543}{96}+\frac {\sqrt {2}\,1155505{}\mathrm {i}}{384}\right )}\right )\,\sqrt {-57873+\sqrt {2}\,23907{}\mathrm {i}}\,1{}\mathrm {i}}{48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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