Optimal. Leaf size=30 \[ -e^{2+2 x+\frac {2 (1+x)}{3}}-(-4+x)^2+\frac {e}{\log (x)} \]
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Rubi [A] time = 0.31, antiderivative size = 28, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {12, 6688, 2194, 2302, 30} \begin {gather*} -x^2+8 x-e^{\frac {8 x}{3}+\frac {8}{3}}+\frac {e}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2194
Rule 2302
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-3 e+\left (24 x-8 e^{\frac {2}{3} (4+4 x)} x-6 x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (24-8 e^{\frac {8}{3}+\frac {8 x}{3}}-6 x-\frac {3 e}{x \log ^2(x)}\right ) \, dx\\ &=8 x-x^2-\frac {8}{3} \int e^{\frac {8}{3}+\frac {8 x}{3}} \, dx-e \int \frac {1}{x \log ^2(x)} \, dx\\ &=-e^{\frac {8}{3}+\frac {8 x}{3}}+8 x-x^2-e \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=-e^{\frac {8}{3}+\frac {8 x}{3}}+8 x-x^2+\frac {e}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 28, normalized size = 0.93 \begin {gather*} -e^{\frac {8}{3}+\frac {8 x}{3}}+8 x-x^2+\frac {e}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 27, normalized size = 0.90 \begin {gather*} -\frac {{\left (x^{2} - 8 \, x + e^{\left (\frac {8}{3} \, x + \frac {8}{3}\right )}\right )} \log \relax (x) - e}{\log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 49.75, size = 31, normalized size = 1.03 \begin {gather*} -\frac {x^{2} \log \relax (x) - 8 \, x \log \relax (x) + e^{\left (\frac {8}{3} \, x + \frac {8}{3}\right )} \log \relax (x) - e}{\log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 0.83
method | result | size |
risch | \(-x^{2}+8 x -{\mathrm e}^{\frac {8 x}{3}+\frac {8}{3}}+\frac {{\mathrm e}}{\ln \relax (x )}\) | \(25\) |
default | \(-x^{2}+8 x -{\mathrm e}^{\frac {8 x}{3}+\frac {8}{3}}+\frac {{\mathrm e}}{\ln \relax (x )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 0.80 \begin {gather*} -x^{2} + 8 \, x + \frac {e}{\log \relax (x)} - e^{\left (\frac {8}{3} \, x + \frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.38, size = 24, normalized size = 0.80 \begin {gather*} 8\,x-{\mathrm {e}}^{\frac {8\,x}{3}+\frac {8}{3}}+\frac {\mathrm {e}}{\ln \relax (x)}-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 22, normalized size = 0.73 \begin {gather*} - x^{2} + 8 x - e^{\frac {8 x}{3} + \frac {8}{3}} + \frac {e}{\log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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