Integrand size = 72, antiderivative size = 15 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x}{3-e+\frac {1}{x^2}+2 x} \]
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Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(15)=30\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 1607, 6820, 2127, 1602} \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=-\frac {(3-e) x^2}{2 \left (2 x^3+(3-e) x^2+1\right )}-\frac {1}{2 \left (2 x^3+(3-e) x^2+1\right )} \]
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Rule 6
Rule 1602
Rule 1607
Rule 2127
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 x^2+(3-e) x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx \\ & = \int \frac {3 x^2+(3-e) x^4}{1+6 x^2+4 x^3+\left (9+e^2\right ) x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx \\ & = \int \frac {x^2 \left (3+(3-e) x^2\right )}{1+6 x^2+4 x^3+\left (9+e^2\right ) x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx \\ & = \int \frac {x^2 \left (3-(-3+e) x^2\right )}{\left (1+(3-e) x^2+2 x^3\right )^2} \, dx \\ & = -\frac {(3-e) x^2}{2 \left (1+(3-e) x^2+2 x^3\right )}-\frac {1}{2} \int \frac {-2 (3-e) x-6 x^2}{\left (1+(3-e) x^2+2 x^3\right )^2} \, dx \\ & = -\frac {1}{2 \left (1+(3-e) x^2+2 x^3\right )}-\frac {(3-e) x^2}{2 \left (1+(3-e) x^2+2 x^3\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {-1+(-3+e) x^2}{2-2 (-3+e) x^2+4 x^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).
Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27
method | result | size |
norman | \(\frac {\left (\frac {3}{2}-\frac {{\mathrm e}}{2}\right ) x^{2}+\frac {1}{2}}{x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1}\) | \(34\) |
risch | \(\frac {\left (\frac {3}{2}-\frac {{\mathrm e}}{2}\right ) x^{2}+\frac {1}{2}}{x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1}\) | \(34\) |
gosper | \(-\frac {x^{2} {\mathrm e}-3 x^{2}-1}{2 \left (x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1\right )}\) | \(36\) |
parallelrisch | \(-\frac {x^{2} {\mathrm e}-3 x^{2}-1}{2 \left (x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1\right )}\) | \(36\) |
default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+4 \textit {\_Z}^{6}+\left (-4 \,{\mathrm e}+12\right ) \textit {\_Z}^{5}+\left ({\mathrm e}^{2}-6 \,{\mathrm e}+9\right ) \textit {\_Z}^{4}+4 \textit {\_Z}^{3}+\left (-2 \,{\mathrm e}+6\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\left (\left ({\mathrm e}-3\right ) \textit {\_R}^{4}-3 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} {\mathrm e}^{2}-5 \textit {\_R}^{4} {\mathrm e}+6 \textit {\_R}^{5}-6 \textit {\_R}^{3} {\mathrm e}+15 \textit {\_R}^{4}+9 \textit {\_R}^{3}-\textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}+3 \textit {\_R}}\right )}{4}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.40 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x^{2} e - 3 \, x^{2} - 1}{2 \, {\left (2 \, x^{3} - x^{2} e + 3 \, x^{2} + 1\right )}} \]
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Time = 0.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x^{2} \left (-3 + e\right ) - 1}{4 x^{3} + x^{2} \cdot \left (6 - 2 e\right ) + 2} \]
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none
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.00 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x^{2} {\left (e - 3\right )} - 1}{2 \, {\left (2 \, x^{3} - x^{2} {\left (e - 3\right )} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.40 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x^{2} e - 3 \, x^{2} - 1}{2 \, {\left (2 \, x^{3} - x^{2} e + 3 \, x^{2} + 1\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {3 x^2+3 x^4-e x^4}{1+6 x^2+4 x^3+9 x^4+e^2 x^4+12 x^5+4 x^6+e \left (-2 x^2-6 x^4-4 x^5\right )} \, dx=\frac {x^2\,\left (\frac {\mathrm {e}}{2}-\frac {3}{2}\right )-\frac {1}{2}}{2\,x^3+\left (3-\mathrm {e}\right )\,x^2+1} \]
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