Optimal. Leaf size=24 \[ x+\log \left (\frac {5}{3} (-4+x)-x+\frac {1}{3+x+e x^2}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps
used = 4, number of rules used = 3, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2099, 642,
1601} \begin {gather*} -\log \left (e x^2+x+3\right )+\log \left (-2 e x^3-2 (1-10 e) x^2+14 x+57\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1601
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-1-2 e x}{3+x+e x^2}+\frac {2 \left (7-2 (1-10 e) x-3 e x^2\right )}{57+14 x-2 (1-10 e) x^2-2 e x^3}\right ) \, dx\\ &=x+2 \int \frac {7-2 (1-10 e) x-3 e x^2}{57+14 x-2 (1-10 e) x^2-2 e x^3} \, dx+\int \frac {-1-2 e x}{3+x+e x^2} \, dx\\ &=x-\log \left (3+x+e x^2\right )+\log \left (57+14 x-2 (1-10 e) x^2-2 e x^3\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 36, normalized size = 1.50 \begin {gather*} x-\log \left (3+x+e x^2\right )+\log \left (57+14 x-2 x^2+20 e x^2-2 e x^3\right ) \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 = 0.42, size = 140, normalized size = 5.83
method | result | size |
norman | \(x -\ln \left (x +3+x^{2} {\mathrm e}\right )+\ln \left (2 x^{3} {\mathrm e}-20 x^{2} {\mathrm e}+2 x^{2}-14 x -57\right )\) | \(40\) |
risch | \(x -\ln \left (-x^{2} {\mathrm e}-x -3\right )+\ln \left (2 x^{3} {\mathrm e}+\left (-20 \,{\mathrm e}+2\right ) x^{2}-14 x -57\right )\) | \(41\) |
default | \(x +\left (\munderset {\textit {\_R} =\RootOf \left (-171+2 \,{\mathrm e}^{2} \textit {\_Z}^{5}+\left (-20 \,{\mathrm e}^{2}+4 \,{\mathrm e}\right ) \textit {\_Z}^{4}+\left (-28 \,{\mathrm e}+2\right ) \textit {\_Z}^{3}+\left (-117 \,{\mathrm e}-8\right ) \textit {\_Z}^{2}-99 \textit {\_Z} \right )}{\sum }\frac {\left (15+2 \textit {\_R}^{4} {\mathrm e}^{2}+4 \textit {\_R}^{3} {\mathrm e}+2 \left (1+6 \,{\mathrm e}\right ) \textit {\_R}^{2}+6 \left (2-{\mathrm e}\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{10 \textit {\_R}^{4} {\mathrm e}^{2}-80 \textit {\_R}^{3} {\mathrm e}^{2}+16 \textit {\_R}^{3} {\mathrm e}-84 \textit {\_R}^{2} {\mathrm e}-234 \textit {\_R} \,{\mathrm e}+6 \textit {\_R}^{2}-16 \textit {\_R} -99}\right )\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 1.58 \begin {gather*} x + \log \left (2 \, x^{3} e - 2 \, x^{2} {\left (10 \, e - 1\right )} - 14 \, x - 57\right ) - \log \left (x^{2} e + x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 38, normalized size = 1.58 \begin {gather*} x - \log \left (x^{2} e + x + 3\right ) + \log \left (2 \, x^{2} + 2 \, {\left (x^{3} - 10 \, x^{2}\right )} e - 14 \, x - 57\right ) \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. 48 vs.
\(2 (22) = 44\).
time = 2.17, size = 48, normalized size = 2.00 \begin {gather*} x - \log {\left (x^{2} + \frac {x}{e} + \frac {3}{e} \right )} + \log {\left (x^{3} + \frac {x^{2} \cdot \left (1 - 10 e\right )}{e} - \frac {7 x}{e} - \frac {57}{2 e} \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. 40 vs.
\(2 (19) = 38\).
time = 0.44, size = 40, normalized size = 1.67 \begin {gather*} x - \log \left (x^{2} e + x + 3\right ) + \log \left ({\left | 2 \, x^{3} e - 20 \, x^{2} e + 2 \, x^{2} - 14 \, x - 57 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 42, normalized size = 1.75 \begin {gather*} x+\ln \left (x^2\,{\mathrm {e}}^{-1}-7\,x\,{\mathrm {e}}^{-1}-\frac {57\,{\mathrm {e}}^{-1}}{2}-10\,x^2+x^3\right )-\ln \left (x^2+{\mathrm {e}}^{-1}\,x+3\,{\mathrm {e}}^{-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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