Optimal. Leaf size=15 \[ x^2+e^x x \log \left (-e^x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.87, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2176, 2194, 2554} \begin {gather*} x^2-e^x \log \left (-e^x\right )+e^x (x+1) \log \left (-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2554
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x^2+\int e^x x \, dx+\int e^x (1+x) \log \left (-e^x\right ) \, dx\\ &=e^x x+x^2-e^x \log \left (-e^x\right )+e^x (1+x) \log \left (-e^x\right )-\int e^x \, dx-\int e^x x \, dx\\ &=-e^x+x^2-e^x \log \left (-e^x\right )+e^x (1+x) \log \left (-e^x\right )+\int e^x \, dx\\ &=x^2-e^x \log \left (-e^x\right )+e^x (1+x) \log \left (-e^x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 14, normalized size = 0.93 \begin {gather*} x \left (x+e^x \log \left (-e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [C] time = 0.57, size = 15, normalized size = 1.00 \begin {gather*} x^{2} + {\left (i \, \pi x + x^{2}\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 13, normalized size = 0.87 \begin {gather*} x e^{x} \log \left (-e^{x}\right ) + x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 28, normalized size = 1.87
method | result | size |
norman | \(-\ln \left (-{\mathrm e}^{x}\right )^{2}+2 x \ln \left (-{\mathrm e}^{x}\right )+\ln \left (-{\mathrm e}^{x}\right ) x \,{\mathrm e}^{x}\) | \(28\) |
default | \({\mathrm e}^{x} x^{2}+{\mathrm e}^{x} \left (\ln \left (-{\mathrm e}^{x}\right )-x \right )+\left (\ln \left (-{\mathrm e}^{x}\right )-x \right ) \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+x^{2}\) | \(42\) |
risch | \(i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{3} x -i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} x +i \pi \,{\mathrm e}^{x} x +x \,{\mathrm e}^{x} \ln \left ({\mathrm e}^{x}\right )+x^{2}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 13, normalized size = 0.87 \begin {gather*} x e^{x} \log \left (-e^{x}\right ) + x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 1.13 \begin {gather*} x^2\,{\mathrm {e}}^x+x^2+\pi \,x\,{\mathrm {e}}^x\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.13, size = 15, normalized size = 1.00 \begin {gather*} x^{2} - \left (- x^{2} - i \pi x\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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