Optimal. Leaf size=20 \[ \frac {-4+x \log (x)}{e}+\log \left (4-x+x^2\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps
used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 6860, 1671,
642, 2332} \begin {gather*} \log \left (x^2-x+4\right )+\frac {x \log (x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 642
Rule 1671
Rule 2332
Rule 6860
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {4-x+x^2+e (-1+2 x)+\left (4-x+x^2\right ) \log (x)}{4-x+x^2} \, dx}{e}\\ &=\frac {\int \left (\frac {4-e-(1-2 e) x+x^2}{4-x+x^2}+\log (x)\right ) \, dx}{e}\\ &=\frac {\int \frac {4-e-(1-2 e) x+x^2}{4-x+x^2} \, dx}{e}+\frac {\int \log (x) \, dx}{e}\\ &=-\frac {x}{e}+\frac {x \log (x)}{e}+\frac {\int \left (1-\frac {e-2 e x}{4-x+x^2}\right ) \, dx}{e}\\ &=\frac {x \log (x)}{e}-\frac {\int \frac {e-2 e x}{4-x+x^2} \, dx}{e}\\ &=\frac {x \log (x)}{e}+\log \left (4-x+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \log (x)+e \log \left (4-x+x^2\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.36, size = 23, normalized size = 1.15
method | result | size |
risch | \(x \,{\mathrm e}^{-1} \ln \left (x \right )+\ln \left (x^{2}-x +4\right )\) | \(17\) |
norman | \(x \,{\mathrm e}^{-1} \ln \left (x \right )+\ln \left (x^{2}-x +4\right )\) | \(19\) |
default | \({\mathrm e}^{-1} \left ({\mathrm e} \ln \left (x^{2}-x +4\right )+x \ln \left (x \right )\right )\) | \(23\) |
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. 60 vs.
\(2 (19) = 38\).
time = 0.64, size = 60, normalized size = 3.00 \begin {gather*} -\frac {1}{15} \, {\left (2 \, \sqrt {15} \arctan \left (\frac {1}{15} \, \sqrt {15} {\left (2 \, x - 1\right )}\right ) e - {\left (2 \, \sqrt {15} \arctan \left (\frac {1}{15} \, \sqrt {15} {\left (2 \, x - 1\right )}\right ) + 15 \, \log \left (x^{2} - x + 4\right )\right )} e - 15 \, x \log \left (x\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} {\left (e \log \left (x^{2} - x + 4\right ) + x \log \left (x\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 15, normalized size = 0.75 \begin {gather*} \frac {x \log {\left (x \right )}}{e} + \log {\left (x^{2} - x + 4 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 20, normalized size = 1.00 \begin {gather*} {\left (e \log \left (x^{2} - x + 4\right ) + x \log \left (x\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 16, normalized size = 0.80 \begin {gather*} \ln \left (x^2-x+4\right )+x\,{\mathrm {e}}^{-1}\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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