Optimal. Leaf size=27 \[ \frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \]
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Rubi [F]
time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx &=3 \int \frac {-47+228 x+120 x^2+19 x^3}{\left (3+x+x^4\right )^4} \, dx+30 \int \frac {x}{\left (3+x+x^4\right )^2} \, dx+\int \frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac {19}{4 \left (3+x+x^4\right )^3}+\frac {1}{\left (3+x+x^4\right )^2}+\frac {1}{4} \int \frac {176-1280 x-300 x^2}{\left (3+x+x^4\right )^3} \, dx+\frac {3}{4} \int \frac {-207+912 x+480 x^2}{\left (3+x+x^4\right )^4} \, dx+30 \int \frac {x}{\left (3+x+x^4\right )^2} \, dx\\ &=-\frac {19}{4 \left (3+x+x^4\right )^3}+\frac {1}{\left (3+x+x^4\right )^2}+\frac {1}{4} \int \left (\frac {176}{\left (3+x+x^4\right )^3}-\frac {1280 x}{\left (3+x+x^4\right )^3}-\frac {300 x^2}{\left (3+x+x^4\right )^3}\right ) \, dx+\frac {3}{4} \int \left (-\frac {207}{\left (3+x+x^4\right )^4}+\frac {912 x}{\left (3+x+x^4\right )^4}+\frac {480 x^2}{\left (3+x+x^4\right )^4}\right ) \, dx+30 \int \frac {x}{\left (3+x+x^4\right )^2} \, dx\\ &=-\frac {19}{4 \left (3+x+x^4\right )^3}+\frac {1}{\left (3+x+x^4\right )^2}+30 \int \frac {x}{\left (3+x+x^4\right )^2} \, dx+44 \int \frac {1}{\left (3+x+x^4\right )^3} \, dx-75 \int \frac {x^2}{\left (3+x+x^4\right )^3} \, dx-\frac {621}{4} \int \frac {1}{\left (3+x+x^4\right )^4} \, dx-320 \int \frac {x}{\left (3+x+x^4\right )^3} \, dx+360 \int \frac {x^2}{\left (3+x+x^4\right )^4} \, dx+684 \int \frac {x}{\left (3+x+x^4\right )^4} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.06, size = 250, normalized size = 9.26
method | result | size |
norman | \(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\) | \(28\) |
gosper | \(-\frac {5 x^{6}-x^{4}-5 x^{2}+3 x -2}{\left (x^{4}+x +3\right )^{3}}\) | \(31\) |
default | \(\frac {\frac {377432}{195075} x^{7}-\frac {1404328}{195075} x^{6}+\frac {234517}{195075} x^{5}+\frac {660506}{195075} x^{4}-\frac {208792}{195075} x^{3}-\frac {13339729}{390150} x^{2}+\frac {89881}{13005} x +\frac {121303}{21675}}{\left (x^{4}+x +3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (377432 \textit {\_R}^{2}-2808656 \textit {\_R} +703551\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}+\frac {-\frac {32}{51} x^{3}+\frac {128}{51} x^{2}-\frac {2}{51} x -\frac {8}{17}}{x^{4}+x +3}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-16 \textit {\_R}^{2}+128 \textit {\_R} -3\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{51}+\frac {-\frac {255032}{195075} x^{11}+\frac {914728}{195075} x^{10}-\frac {226867}{195075} x^{9}-\frac {701338}{195075} x^{8}+\frac {236024}{195075} x^{7}+\frac {13501313}{390150} x^{6}-\frac {2360372}{195075} x^{5}-\frac {1873778}{195075} x^{4}+\frac {10935781}{390150} x^{3}+\frac {3415123}{43350} x^{2}-\frac {188961}{7225} x -\frac {76253}{7225}}{\left (x^{4}+x +3\right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-255032 \textit {\_R}^{2}+1829456 \textit {\_R} -680601\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}\) | \(250\) |
risch | \(\frac {\frac {377432}{195075} x^{7}-\frac {1404328}{195075} x^{6}+\frac {234517}{195075} x^{5}+\frac {660506}{195075} x^{4}-\frac {208792}{195075} x^{3}-\frac {13339729}{390150} x^{2}+\frac {89881}{13005} x +\frac {121303}{21675}}{\left (x^{4}+x +3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (377432 \textit {\_R}^{2}-2808656 \textit {\_R} +703551\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}+\frac {-\frac {32}{51} x^{3}+\frac {128}{51} x^{2}-\frac {2}{51} x -\frac {8}{17}}{x^{4}+x +3}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-16 \textit {\_R}^{2}+128 \textit {\_R} -3\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{51}+\frac {-\frac {255032}{195075} x^{11}+\frac {914728}{195075} x^{10}-\frac {226867}{195075} x^{9}-\frac {701338}{195075} x^{8}+\frac {236024}{195075} x^{7}+\frac {13501313}{390150} x^{6}-\frac {2360372}{195075} x^{5}-\frac {1873778}{195075} x^{4}+\frac {10935781}{390150} x^{3}+\frac {3415123}{43350} x^{2}-\frac {188961}{7225} x -\frac {76253}{7225}}{\left (x^{4}+x +3\right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-255032 \textit {\_R}^{2}+1829456 \textit {\_R} -680601\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (30) = 60\).
time = 0.29, size = 65, normalized size = 2.41 \begin {gather*} -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \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. 65 vs.
\(2 (30) = 60\).
time = 0.38, size = 65, normalized size = 2.41 \begin {gather*} -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (26) = 52\).
time = 0.15, size = 60, normalized size = 2.22 \begin {gather*} \frac {- 5 x^{6} + x^{4} + 5 x^{2} - 3 x + 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \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. 197 vs.
\(2 (30) = 60\).
time = 5.46, size = 197, normalized size = 7.30 \begin {gather*} \frac {1}{195075} \, x {\left (\frac {377432 \, x^{2} - 2808656 \, x + 703551}{x^{4} + x + 3} - \frac {255032 \, x^{2} - 1829456 \, x + 680601}{x^{4} + x + 3} - \frac {7650 \, {\left (16 \, x^{2} - 128 \, x + 3\right )}}{x^{4} + x + 3}\right )} - \frac {2 \, {\left (16 \, x^{3} - 64 \, x^{2} + x + 12\right )}}{51 \, {\left (x^{4} + x + 3\right )}} + \frac {754864 \, x^{7} - 2808656 \, x^{6} + 469034 \, x^{5} + 1321012 \, x^{4} - 417584 \, x^{3} - 13339729 \, x^{2} + 2696430 \, x + 2183454}{390150 \, {\left (x^{4} + x + 3\right )}^{2}} - \frac {510064 \, x^{11} - 1829456 \, x^{10} + 453734 \, x^{9} + 1402676 \, x^{8} - 472048 \, x^{7} - 13501313 \, x^{6} + 4720744 \, x^{5} + 3747556 \, x^{4} - 10935781 \, x^{3} - 30736107 \, x^{2} + 10203894 \, x + 4117662}{390150 \, {\left (x^{4} + x + 3\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 27, normalized size = 1.00 \begin {gather*} \frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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