Optimal. Leaf size=33 \[ \frac {11}{3}-e^{2 x}-x+(1-x) x+\frac {x}{2+\frac {\log (x)}{x}} \]
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Rubi [F]
time = 0.66, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-x+2 x^2-8 e^{2 x} x^2-8 x^3+\left (2 x-8 e^{2 x} x-8 x^2\right ) \log (x)+\left (-2 e^{2 x}-2 x\right ) \log ^2(x)}{4 x^2+4 x \log (x)+\log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+2 x^2-8 e^{2 x} x^2-8 x^3+\left (2 x-8 e^{2 x} x-8 x^2\right ) \log (x)+\left (-2 e^{2 x}-2 x\right ) \log ^2(x)}{(2 x+\log (x))^2} \, dx\\ &=\int \left (-2 e^{2 x}-\frac {x}{(2 x+\log (x))^2}+\frac {2 x^2}{(2 x+\log (x))^2}-\frac {8 x^3}{(2 x+\log (x))^2}+\frac {2 x \log (x)}{(2 x+\log (x))^2}-\frac {8 x^2 \log (x)}{(2 x+\log (x))^2}-\frac {2 x \log ^2(x)}{(2 x+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int e^{2 x} \, dx\right )+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \frac {x \log (x)}{(2 x+\log (x))^2} \, dx-2 \int \frac {x \log ^2(x)}{(2 x+\log (x))^2} \, dx-8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-8 \int \frac {x^2 \log (x)}{(2 x+\log (x))^2} \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ &=-e^{2 x}+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \left (-\frac {2 x^2}{(2 x+\log (x))^2}+\frac {x}{2 x+\log (x)}\right ) \, dx-2 \int \left (x+\frac {4 x^3}{(2 x+\log (x))^2}-\frac {4 x^2}{2 x+\log (x)}\right ) \, dx-8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-8 \int \left (-\frac {2 x^3}{(2 x+\log (x))^2}+\frac {x^2}{2 x+\log (x)}\right ) \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ &=-e^{2 x}-x^2+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \frac {x}{2 x+\log (x)} \, dx-4 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-2 \left (8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx\right )+16 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 25, normalized size = 0.76 \begin {gather*} -e^{2 x}-x^2+\frac {x^2}{2 x+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 54, normalized size = 1.64
method | result | size |
risch | \(-x^{2}-{\mathrm e}^{2 x}+\frac {x^{2}}{2 x +\ln \left (x \right )}\) | \(25\) |
default | \(-\frac {x^{2} \ln \left (x \right )-x^{2}+2 x^{3}}{2 x +\ln \left (x \right )}+\frac {-2 x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} \ln \left (x \right )}{2 x +\ln \left (x \right )}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2 \, x^{3} + x^{2} \log \left (x\right ) - x^{2} + {\left (2 \, x + \log \left (x\right )\right )} e^{\left (2 \, x\right )}}{2 \, x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 39, normalized size = 1.18 \begin {gather*} -\frac {2 \, x^{3} - x^{2} + 2 \, x e^{\left (2 \, x\right )} + {\left (x^{2} + e^{\left (2 \, x\right )}\right )} \log \left (x\right )}{2 \, x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 17, normalized size = 0.52 \begin {gather*} - x^{2} + \frac {x^{2}}{2 x + \log {\left (x \right )}} - e^{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 41, normalized size = 1.24 \begin {gather*} -\frac {2 \, x^{3} + x^{2} \log \left (x\right ) - x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (x\right )}{2 \, x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 28, normalized size = 0.85 \begin {gather*} \frac {x^3}{x\,\ln \left (x\right )+2\,x^2}-x^2-{\mathrm {e}}^{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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