Optimal. Leaf size=19 \[ \left (-e^x+e^{e^{2 x}+x}\right ) x^2 \]
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Rubi [B] time = 0.10, antiderivative size = 44, normalized size of antiderivative = 2.32, number of steps used = 10, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {1593, 2196, 2176, 2194, 2288} \begin {gather*} \frac {e^{x+e^{2 x}} \left (2 e^{2 x} x^2+x^2\right )}{2 e^{2 x}+1}-e^x x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^x \left (-2 x-x^2\right ) \, dx+\int e^{e^{2 x}+x} \left (2 x+x^2+2 e^{2 x} x^2\right ) \, dx\\ &=\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}+\int e^x (-2-x) x \, dx\\ &=\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}+\int \left (-2 e^x x-e^x x^2\right ) \, dx\\ &=\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}-2 \int e^x x \, dx-\int e^x x^2 \, dx\\ &=-2 e^x x-e^x x^2+\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}+2 \int e^x \, dx+2 \int e^x x \, dx\\ &=2 e^x-e^x x^2+\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}-2 \int e^x \, dx\\ &=-e^x x^2+\frac {e^{e^{2 x}+x} \left (x^2+2 e^{2 x} x^2\right )}{1+2 e^{2 x}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 16, normalized size = 0.84 \begin {gather*} e^x \left (-1+e^{e^{2 x}}\right ) x^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 19, normalized size = 1.00 \begin {gather*} x^{2} e^{\left (x + e^{\left (2 \, x\right )}\right )} - x^{2} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 19, normalized size = 1.00 \begin {gather*} x^{2} e^{\left (x + e^{\left (2 \, x\right )}\right )} - x^{2} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 1.05
method | result | size |
default | \({\mathrm e}^{{\mathrm e}^{2 x}+x} x^{2}-{\mathrm e}^{x} x^{2}\) | \(20\) |
norman | \({\mathrm e}^{{\mathrm e}^{2 x}+x} x^{2}-{\mathrm e}^{x} x^{2}\) | \(20\) |
risch | \({\mathrm e}^{{\mathrm e}^{2 x}+x} x^{2}-{\mathrm e}^{x} x^{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 31, normalized size = 1.63 \begin {gather*} x^{2} e^{\left (x + e^{\left (2 \, x\right )}\right )} - {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 2 \, {\left (x - 1\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.71, size = 13, normalized size = 0.68 \begin {gather*} x^2\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{{\mathrm {e}}^{2\,x}}-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 17, normalized size = 0.89 \begin {gather*} - x^{2} e^{x} + x^{2} e^{x + e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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