Optimal. Leaf size=27 \[ \frac {\log ^2(4)}{x^2}+4 x \left (e^x+2 x-\log (x)\right )-\log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps
used = 9, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 2207, 2225,
2332} \begin {gather*} 8 x^2+\frac {\log ^2(4)}{x^2}-4 e^x+4 e^x (x+1)-4 x \log (x)-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2207
Rule 2225
Rule 2332
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4 e^x (1+x)+\frac {-x^2-4 x^3+16 x^4-2 \log ^2(4)-4 x^3 \log (x)}{x^3}\right ) \, dx\\ &=4 \int e^x (1+x) \, dx+\int \frac {-x^2-4 x^3+16 x^4-2 \log ^2(4)-4 x^3 \log (x)}{x^3} \, dx\\ &=4 e^x (1+x)-4 \int e^x \, dx+\int \left (\frac {-x^2-4 x^3+16 x^4-2 \log ^2(4)}{x^3}-4 \log (x)\right ) \, dx\\ &=-4 e^x+4 e^x (1+x)-4 \int \log (x) \, dx+\int \frac {-x^2-4 x^3+16 x^4-2 \log ^2(4)}{x^3} \, dx\\ &=-4 e^x+4 x+4 e^x (1+x)-4 x \log (x)+\int \left (-4-\frac {1}{x}+16 x-\frac {2 \log ^2(4)}{x^3}\right ) \, dx\\ &=-4 e^x+8 x^2+4 e^x (1+x)+\frac {\log ^2(4)}{x^2}-\log (x)-4 x \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 29, normalized size = 1.07 \begin {gather*} 4 e^x x+8 x^2+\frac {\log ^2(4)}{x^2}-\log (x)-4 x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 30, normalized size = 1.11
method | result | size |
default | \(4 \,{\mathrm e}^{x} x +8 x^{2}-\ln \left (x \right )+\frac {4 \ln \left (2\right )^{2}}{x^{2}}-4 x \ln \left (x \right )\) | \(30\) |
risch | \(-4 x \ln \left (x \right )-\frac {-8 x^{4}-4 \,{\mathrm e}^{x} x^{3}+x^{2} \ln \left (x \right )-4 \ln \left (2\right )^{2}}{x^{2}}\) | \(37\) |
norman | \(\frac {-x^{2} \ln \left (x \right )+8 x^{4}+4 \ln \left (2\right )^{2}-4 x^{3} \ln \left (x \right )+4 \,{\mathrm e}^{x} x^{3}}{x^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 1.30 \begin {gather*} 8 \, x^{2} + 4 \, {\left (x - 1\right )} e^{x} - 4 \, x \log \left (x\right ) + \frac {4 \, \log \left (2\right )^{2}}{x^{2}} + 4 \, e^{x} - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 36, normalized size = 1.33 \begin {gather*} \frac {8 \, x^{4} + 4 \, x^{3} e^{x} + 4 \, \log \left (2\right )^{2} - {\left (4 \, x^{3} + x^{2}\right )} \log \left (x\right )}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 31, normalized size = 1.15 \begin {gather*} 8 x^{2} + 4 x e^{x} - 4 x \log {\left (x \right )} - \log {\left (x \right )} + \frac {4 \log {\left (2 \right )}^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 37, normalized size = 1.37 \begin {gather*} \frac {8 \, x^{4} + 4 \, x^{3} e^{x} - 4 \, x^{3} \log \left (x\right ) - x^{2} \log \left (x\right ) + 4 \, \log \left (2\right )^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 30, normalized size = 1.11 \begin {gather*} \frac {4\,{\ln \left (2\right )}^2}{x^2}-\ln \left (x\right )+x\,\left (4\,{\mathrm {e}}^x-4\,\ln \left (x\right )\right )+8\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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