Optimal. Leaf size=232 \[ -17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 232, 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}{2} \left (14395+26499 \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}{2} \left (14395+26499 \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 {5 x^3}{3}-\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (26499 \sqrt {3}-14395\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {25 \left (x^2+3\right ) x}{8 \left (x^4+2 x^2+3\right )}-17 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^4 \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 (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \frac {450-150 x^2-336 x^4+240 x^6}{3+2 x^2+x^4} \, dx\\ &=-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{48} \int \left (-816+240 x^2+\frac {6 \left (483+127 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{8} \int \frac {483+127 x^2}{3+2 x^2+x^4} \, dx\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {483 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (483-127 \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 {483 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (483-127 \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 )}}\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{32} \left (127+161 \sqrt {3}\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx+\frac {1}{32} \left (127+161 \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}{2} \left (-14395+26499 \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}{2} \left (-14395+26499 \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\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{16} \left (-127-161 \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}{16} \left (-127-161 \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 )\\ &=-17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {1}{16} \sqrt {\frac {1}{2} \left (14395+26499 \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}{2} \left (14395+26499 \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}{2} \left (-14395+26499 \sqrt {3}\right )} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (-14395+26499 \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 = 129, normalized size = 0.56 \begin {gather*} -17 x+\frac {5 x^3}{3}-\frac {25 x \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {\left (-356 i+127 \sqrt {2}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{16 \sqrt {2-2 i \sqrt {2}}}+\frac {\left (356 i+127 \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 = 285, normalized size = 1.23
method | result | size |
risch | \(\frac {5 x^{3}}{3}-17 x +\frac {-\frac {25}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\left (127 \textit {\_R}^{2}+483\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{32}\) | \(69\) |
default | \(\frac {5 x^{3}}{3}-17 x +\frac {-\frac {25}{8} x^{3}-\frac {75}{8} x}{x^{4}+2 x^{2}+3}+\frac {\left (-17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-178 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}-x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {\left (322 \sqrt {3}+\frac {\left (-17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-178 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+178 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+\sqrt {3}+x \sqrt {-2+2 \sqrt {3}}\right )}{64}+\frac {\left (322 \sqrt {3}-\frac {\left (17 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+178 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}\) | \(285\) |
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. 510 vs.
\(2 (163) = 326\).
time = 0.37, size = 510, normalized size = 2.20 \begin {gather*} \frac {2159655360 \, x^{7} - 17709173952 \, x^{5} - 123268 \cdot 143883^{\frac {1}{4}} \sqrt {219} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {14395 \, \sqrt {3} + 79497} \arctan \left (\frac {1}{59850021621146706} \cdot 143883^{\frac {3}{4}} \sqrt {30817} \sqrt {219} \sqrt {73} \sqrt {143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 20246769 \, x^{2} + 20246769 \, \sqrt {3}} {\left (161 \, \sqrt {3} \sqrt {2} - 127 \, \sqrt {2}\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{8868084822} \cdot 143883^{\frac {3}{4}} \sqrt {219} {\left (161 \, \sqrt {3} \sqrt {2} x - 127 \, \sqrt {2} x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - 123268 \cdot 143883^{\frac {1}{4}} \sqrt {219} \sqrt {3} \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \sqrt {14395 \, \sqrt {3} + 79497} \arctan \left (\frac {1}{7241852616158751426} \cdot 143883^{\frac {3}{4}} \sqrt {30817} \sqrt {219} \sqrt {-1068793 \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 21639604979817 \, x^{2} + 21639604979817 \, \sqrt {3}} {\left (161 \, \sqrt {3} \sqrt {2} - 127 \, \sqrt {2}\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{8868084822} \cdot 143883^{\frac {3}{4}} \sqrt {219} {\left (161 \, \sqrt {3} \sqrt {2} x - 127 \, \sqrt {2} x\right )} \sqrt {14395 \, \sqrt {3} + 79497} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) - 143883^{\frac {1}{4}} \sqrt {219} {\left (79497 \, x^{4} + 158994 \, x^{2} - 14395 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 238491\right )} \sqrt {14395 \, \sqrt {3} + 79497} \log \left (\frac {1068793}{30817} \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 702197001 \, x^{2} + 702197001 \, \sqrt {3}\right ) + 143883^{\frac {1}{4}} \sqrt {219} {\left (79497 \, x^{4} + 158994 \, x^{2} - 14395 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 3\right )} + 238491\right )} \sqrt {14395 \, \sqrt {3} + 79497} \log \left (-\frac {1068793}{30817} \cdot 143883^{\frac {1}{4}} \sqrt {219} {\left (127 \, \sqrt {3} x - 483 \, x\right )} \sqrt {14395 \, \sqrt {3} + 79497} + 702197001 \, x^{2} + 702197001 \, \sqrt {3}\right ) - 41627357064 \, x^{3} - 78233515416 \, x}{1295793216 \, {\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 = 0.34, size = 60, normalized size = 0.26 \begin {gather*} \frac {5 x^{3}}{3} - 17 x + \frac {- 25 x^{3} - 75 x}{8 x^{4} + 16 x^{2} + 24} + \operatorname {RootSum} {\left (1048576 t^{4} + 29480960 t^{2} + 2106591003, \left ( t \mapsto t \log {\left (\frac {557056 t^{3}}{816619683} + \frac {166600064 t}{816619683} + 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. 573 vs.
\(2 (163) = 326\).
time = 3.69, size = 573, normalized size = 2.47 \begin {gather*} \frac {5}{3} \, x^{3} - \frac {1}{20736} \, \sqrt {2} {\left (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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}{20736} \, \sqrt {2} {\left (127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 2286 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 127 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 17388 \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}{41472} \, \sqrt {2} {\left (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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}{41472} \, \sqrt {2} {\left (2286 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 127 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 127 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2286 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 17388 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 17388 \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 ) - 17 \, x - \frac {25 \, {\left (x^{3} + 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 = 0.09, size = 162, normalized size = 0.70 \begin {gather*} \frac {5\,x^3}{3}-\frac {\frac {25\,x^3}{8}+\frac {75\,x}{8}}{x^4+2\,x^2+3}-17\,x+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}-\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (-\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395-\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,30817{}\mathrm {i}}{64\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}+\frac {30817\,\sqrt {2}\,x\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}}{128\,\left (\frac {1571667}{64}+\frac {\sqrt {2}\,14884611{}\mathrm {i}}{128}\right )}\right )\,\sqrt {-14395+\sqrt {2}\,30817{}\mathrm {i}}\,1{}\mathrm {i}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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