Optimal. Leaf size=22 \[ -1+\frac {x}{2}-e^x x-x^2 \log ^2(x) \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 14, 2176, 2194, 2304, 2305} \begin {gather*} -x^2 \log ^2(x)+e^x+\frac {x}{2}-e^x (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rule 2304
Rule 2305
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {x+e^x (-2-2 x) x-4 x^2 \log (x)-4 x^2 \log ^2(x)}{x} \, dx\\ &=\frac {1}{2} \int \left (1-2 e^x (1+x)-4 x \log (x)-4 x \log ^2(x)\right ) \, dx\\ &=\frac {x}{2}-2 \int x \log (x) \, dx-2 \int x \log ^2(x) \, dx-\int e^x (1+x) \, dx\\ &=\frac {x}{2}+\frac {x^2}{2}-e^x (1+x)-x^2 \log (x)-x^2 \log ^2(x)+2 \int x \log (x) \, dx+\int e^x \, dx\\ &=e^x+\frac {x}{2}-e^x (1+x)-x^2 \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{2}-e^x x-x^2 \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 20, normalized size = 0.91 \begin {gather*} -x^{2} \log \relax (x)^{2} + \frac {1}{2} \, x - e^{\left (x + \log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 18, normalized size = 0.82 \begin {gather*} -x^{2} \log \relax (x)^{2} - x e^{x} + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 0.86
method | result | size |
default | \(\frac {x}{2}-x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x} x\) | \(19\) |
risch | \(\frac {x}{2}-x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x} x\) | \(19\) |
norman | \(\frac {x}{2}-x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x +\ln \relax (x )}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 44, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - x^{2} \log \relax (x) + \frac {1}{2} \, x^{2} - {\left (x - 1\right )} e^{x} + \frac {1}{2} \, x - e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.58, size = 16, normalized size = 0.73 \begin {gather*} -\frac {x\,\left (2\,x\,{\ln \relax (x)}^2+2\,{\mathrm {e}}^x-1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 15, normalized size = 0.68 \begin {gather*} - x^{2} \log {\relax (x )}^{2} - x e^{x} + \frac {x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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