Optimal. Leaf size=15 \[ \frac {x}{3-e+\frac {1}{x^2}+2 x} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(15)=30\).
time = 0.25, antiderivative size = 53, normalized size of antiderivative = 3.53, number of steps
used = 6, number of rules used = 5, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {6, 1607, 6820,
2127, 1602} \begin {gather*} -\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 1602
Rule 1607
Rule 2127
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 27, normalized size = 1.80 \begin {gather*} \frac {-1+(-3+e) x^2}{2-2 (-3+e) x^2+4 x^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
3.
time = 2.18, size = 123, normalized size = 8.20
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 | \(-\frac {x^{2} {\mathrm e}-3 x^{2}-1}{2 \left (x^{2} {\mathrm e}-2 x^{3}-3 x^{2}-1\right )}\) | \(36\) |
default | \(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+4 \textit {\_Z}^{6}+\left (-4 \,{\mathrm e}+12\right ) \textit {\_Z}^{5}+\left ({\mathrm e}^{2}-6 \,{\mathrm e}+9\right ) \textit {\_Z}^{4}+4 \textit {\_Z}^{3}+\left (-2 \,{\mathrm e}+6\right ) \textit {\_Z}^{2}\right )}{\sum }\frac {\left (\left ({\mathrm e}-3\right ) \textit {\_R}^{4}-3 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} {\mathrm e}^{2}-5 \textit {\_R}^{4} {\mathrm e}+6 \textit {\_R}^{5}-6 \textit {\_R}^{3} {\mathrm e}+15 \textit {\_R}^{4}+9 \textit {\_R}^{3}-\textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}+3 \textit {\_R}}\right )}{4}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 30, normalized size = 2.00 \begin {gather*} \frac {x^{2} {\left (e - 3\right )} - 1}{2 \, {\left (2 \, x^{3} - x^{2} {\left (e - 3\right )} + 1\right )}} \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. 36 vs.
\(2 (16) = 32\).
time = 0.47, size = 36, normalized size = 2.40 \begin {gather*} \frac {x^{2} e - 3 \, x^{2} - 1}{2 \, {\left (2 \, x^{3} - x^{2} e + 3 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 26, normalized size = 1.73 \begin {gather*} \frac {x^{2} \left (-3 + e\right ) - 1}{4 x^{3} + x^{2} \cdot \left (6 - 2 e\right ) + 2} \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. 36 vs.
\(2 (16) = 32\).
time = 0.41, size = 36, normalized size = 2.40 \begin {gather*} \frac {x^{2} e - 3 \, x^{2} - 1}{2 \, {\left (2 \, x^{3} - x^{2} e + 3 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 31, normalized size = 2.07 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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