Optimal. Leaf size=20 \[ \frac {1}{2} x \left (3 e (-2+x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps
used = 5, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 2369, 2350}
\begin {gather*} \frac {3}{2} e x^2 \log \left (x^3\right )-\frac {1}{2} x \left (6 e-\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2350
Rule 2369
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (e (-18+9 x)+3 \log \left (\frac {5}{4}\right )+\left (e (-6+6 x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right )\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \int \left (e (-6+6 x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \int \left (-6 e+6 e x+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \left (3 e x^2-x \left (6 e-\log \left (\frac {5}{4}\right )\right )\right ) \log \left (x^3\right )-\frac {3}{2} \int \left (3 e (-2+x)+\log \left (\frac {5}{4}\right )\right ) \, dx\\ &=\frac {1}{2} \left (3 e x^2-x \left (6 e-\log \left (\frac {5}{4}\right )\right )\right ) \log \left (x^3\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 1.70 \begin {gather*} -3 e x \log \left (x^3\right )+\frac {3}{2} e x^2 \log \left (x^3\right )+\frac {1}{2} x \log \left (\frac {5}{4}\right ) \log \left (x^3\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. \(79\) vs.
\(2(17)=34\).
time = 0.30, size = 80, normalized size = 4.00
method | result | size |
norman | \(\left (-3 \,{\mathrm e}+\frac {\ln \left (5\right )}{2}-\ln \left (2\right )\right ) x \ln \left (x^{3}\right )+\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}\) | \(32\) |
risch | \(\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}-3 \,{\mathrm e} \ln \left (x^{3}\right ) x -\ln \left (2\right ) \ln \left (x^{3}\right ) x +\frac {\ln \left (5\right ) \ln \left (x^{3}\right ) x}{2}\) | \(40\) |
default | \(\frac {{\mathrm e} \left (\frac {9}{2} x^{2}-18 x \right )}{2}+\frac {\ln \left (5\right ) \ln \left (x^{3}\right ) x}{2}-\frac {3 x \ln \left (5\right )}{2}-3 \,{\mathrm e} \ln \left (x^{3}\right ) x +9 x \,{\mathrm e}-\ln \left (2\right ) \ln \left (x^{3}\right ) x +3 x \ln \left (2\right )+\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}-\frac {9 x^{2} {\mathrm e}}{4}+\frac {3 \ln \left (\frac {5}{4}\right ) x}{2}\) | \(80\) |
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. 58 vs.
\(2 (17) = 34\).
time = 0.27, size = 58, normalized size = 2.90 \begin {gather*} -\frac {9}{4} \, x^{2} e + \frac {3}{2} \, x {\left (6 \, e - \log \left (\frac {5}{4}\right )\right )} + \frac {9}{4} \, {\left (x^{2} - 4 \, x\right )} e + \frac {3}{2} \, x \log \left (\frac {5}{4}\right ) + \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 32, normalized size = 1.60 \begin {gather*} \left (\frac {3 e x^{2}}{2} - 3 e x - x \log {\left (2 \right )} + \frac {x \log {\left (5 \right )}}{2}\right ) \log {\left (x^{3} \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. 46 vs.
\(2 (17) = 34\).
time = 0.41, size = 46, normalized size = 2.30 \begin {gather*} -\frac {9}{4} \, x^{2} e + \frac {9}{4} \, {\left (x^{2} - 4 \, x\right )} e + 9 \, x e + \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.59, size = 19, normalized size = 0.95 \begin {gather*} \frac {x\,\ln \left (x^3\right )\,\left (\ln \left (\frac {5}{4}\right )-6\,\mathrm {e}+3\,x\,\mathrm {e}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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