Optimal. Leaf size=24 \[ x \left (-x+e^{x/3} \left (\left (1+e^2\right )^2+\log (x)\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(24)=48\).
time = 0.08, antiderivative size = 85, normalized size of antiderivative = 3.54, number of steps
used = 8, number of rules used = 5, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {12, 2218, 2207,
2225, 2634} \begin {gather*} -x^2-3 e^{x/3}+e^{x/3} \left (\left (1+e^2\right )^2 x+3 \left (2+2 e^2+e^4\right )\right )-3 \left (1+e^2\right )^2 e^{x/3}-3 e^{x/3} \log (x)+e^{x/3} (x+3) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2218
Rule 2225
Rule 2634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-6 x+e^{x/3} \left (6+x+e^4 (3+x)+e^2 (6+2 x)\right )+e^{x/3} (3+x) \log (x)\right ) \, dx\\ &=-x^2+\frac {1}{3} \int e^{x/3} \left (6+x+e^4 (3+x)+e^2 (6+2 x)\right ) \, dx+\frac {1}{3} \int e^{x/3} (3+x) \log (x) \, dx\\ &=-x^2-3 e^{x/3} \log (x)+e^{x/3} (3+x) \log (x)-\frac {1}{3} \int 3 e^{x/3} \, dx+\frac {1}{3} \int e^{x/3} \left (3 \left (2+2 e^2+e^4\right )+\left (1+e^2\right )^2 x\right ) \, dx\\ &=-x^2+e^{x/3} \left (3 \left (2+2 e^2+e^4\right )+\left (1+e^2\right )^2 x\right )-3 e^{x/3} \log (x)+e^{x/3} (3+x) \log (x)-\left (1+e^2\right )^2 \int e^{x/3} \, dx-\int e^{x/3} \, dx\\ &=-3 e^{x/3}-3 e^{x/3} \left (1+e^2\right )^2-x^2+e^{x/3} \left (3 \left (2+2 e^2+e^4\right )+\left (1+e^2\right )^2 x\right )-3 e^{x/3} \log (x)+e^{x/3} (3+x) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).
time = 0.08, size = 51, normalized size = 2.12 \begin {gather*} \frac {1}{3} \left (-9 e^{x/3}-3 x^2+e^{x/3} \left (9+3 \left (1+e^2\right )^2 x\right )+3 e^{x/3} x \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs.
\(2(20)=40\).
time = 0.23, size = 77, normalized size = 3.21
method | result | size |
norman | \(\left ({\mathrm e}^{4}+2 \,{\mathrm e}^{2}+1\right ) x \,{\mathrm e}^{\frac {x}{3}}+{\mathrm e}^{\frac {x}{3}} \ln \left (x \right ) x -x^{2}\) | \(31\) |
risch | \({\mathrm e}^{\frac {x}{3}} \ln \left (x \right ) x -3 \,{\mathrm e}^{\frac {x}{3}}+\frac {\left (3 x \,{\mathrm e}^{4}+6 \,{\mathrm e}^{2} x +3 x +9\right ) {\mathrm e}^{\frac {x}{3}}}{3}-x^{2}\) | \(42\) |
default | \(-x^{2}+x \,{\mathrm e}^{\frac {x}{3}}+6 \,{\mathrm e}^{2} {\mathrm e}^{\frac {x}{3}}+3 \,{\mathrm e}^{4} {\mathrm e}^{\frac {x}{3}}+3 \,{\mathrm e}^{4} \left (\frac {x \,{\mathrm e}^{\frac {x}{3}}}{3}-{\mathrm e}^{\frac {x}{3}}\right )+6 \,{\mathrm e}^{2} \left (\frac {x \,{\mathrm e}^{\frac {x}{3}}}{3}-{\mathrm e}^{\frac {x}{3}}\right )+{\mathrm e}^{\frac {x}{3}} \ln \left (x \right ) x\) | \(77\) |
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. 73 vs.
\(2 (20) = 40\).
time = 0.30, size = 73, normalized size = 3.04 \begin {gather*} x e^{\left (\frac {1}{3} \, x\right )} \log \left (x\right ) - x^{2} + {\left (x e^{4} - 3 \, e^{4}\right )} e^{\left (\frac {1}{3} \, x\right )} + 2 \, {\left (x e^{2} - 3 \, e^{2}\right )} e^{\left (\frac {1}{3} \, x\right )} + {\left (x - 3\right )} e^{\left (\frac {1}{3} \, x\right )} + 3 \, e^{\left (\frac {1}{3} \, x\right )} + 3 \, e^{\left (\frac {1}{3} \, x + 4\right )} + 6 \, e^{\left (\frac {1}{3} \, x + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 30, normalized size = 1.25 \begin {gather*} x e^{\left (\frac {1}{3} \, x\right )} \log \left (x\right ) - x^{2} + {\left (x e^{4} + 2 \, x e^{2} + x\right )} e^{\left (\frac {1}{3} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 26, normalized size = 1.08 \begin {gather*} - x^{2} + \left (x \log {\left (x \right )} + x + 2 x e^{2} + x e^{4}\right ) e^{\frac {x}{3}} \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. 45 vs.
\(2 (20) = 40\).
time = 0.40, size = 45, normalized size = 1.88 \begin {gather*} x e^{\left (\frac {1}{3} \, x\right )} \log \left (x\right ) - x^{2} + {\left (x + 3\right )} e^{\left (\frac {1}{3} \, x\right )} + x e^{\left (\frac {1}{3} \, x + 4\right )} + 2 \, x e^{\left (\frac {1}{3} \, x + 2\right )} - 3 \, e^{\left (\frac {1}{3} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.23, size = 27, normalized size = 1.12 \begin {gather*} x\,\left ({\mathrm {e}}^{x/3}\,\ln \left (x\right )+{\mathrm {e}}^{x/3}\,{\left ({\mathrm {e}}^2+1\right )}^2\right )-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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