Optimal. Leaf size=29 \[ -\log \left (\frac {5}{2}\right )+e^{e^x} \left (2-\log \left (4+e^{x-x^2}\right )\right ) \]
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Rubi [A] time = 2.03, antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 5, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6688, 6742, 2282, 2194, 2554} \begin {gather*} 2 e^{e^x}-e^{e^x} \log \left (e^{x-x^2}+4\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2194
Rule 2282
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^x+x} \left (-1+2 e^x+8 e^{x^2}+2 x-\left (e^x+4 e^{x^2}\right ) \log \left (4+e^{x-x^2}\right )\right )}{e^x+4 e^{x^2}} \, dx\\ &=\int \left (2 e^{e^x+x}+\frac {e^{e^x+x} (-1+2 x)}{e^x+4 e^{x^2}}-e^{e^x+x} \log \left (4+e^{x-x^2}\right )\right ) \, dx\\ &=2 \int e^{e^x+x} \, dx+\int \frac {e^{e^x+x} (-1+2 x)}{e^x+4 e^{x^2}} \, dx-\int e^{e^x+x} \log \left (4+e^{x-x^2}\right ) \, dx\\ &=-e^{e^x} \log \left (4+e^{x-x^2}\right )+2 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+\int \frac {e^{e^x+x} (1-2 x)}{e^x+4 e^{x^2}} \, dx+\int \left (-\frac {e^{e^x+x}}{e^x+4 e^{x^2}}+\frac {2 e^{e^x+x} x}{e^x+4 e^{x^2}}\right ) \, dx\\ &=2 e^{e^x}-e^{e^x} \log \left (4+e^{x-x^2}\right )+2 \int \frac {e^{e^x+x} x}{e^x+4 e^{x^2}} \, dx-\int \frac {e^{e^x+x}}{e^x+4 e^{x^2}} \, dx+\int \left (\frac {e^{e^x+x}}{e^x+4 e^{x^2}}-\frac {2 e^{e^x+x} x}{e^x+4 e^{x^2}}\right ) \, dx\\ &=2 e^{e^x}-e^{e^x} \log \left (4+e^{x-x^2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.13, size = 21, normalized size = 0.72 \begin {gather*} -e^{e^x} \left (-2+\log \left (4+e^{x-x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 31, normalized size = 1.07 \begin {gather*} -{\left (e^{\left (x + e^{x}\right )} \log \left (e^{\left (-x^{2} + x\right )} + 4\right ) - 2 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 39, normalized size = 1.34 \begin {gather*} {\left (x^{2} e^{\left (x + e^{x}\right )} - e^{\left (x + e^{x}\right )} \log \left (4 \, e^{\left (x^{2}\right )} + e^{x}\right ) + 2 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 22, normalized size = 0.76
method | result | size |
risch | \(-{\mathrm e}^{{\mathrm e}^{x}} \ln \left ({\mathrm e}^{-x \left (x -1\right )}+4\right )+2 \,{\mathrm e}^{{\mathrm e}^{x}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 25, normalized size = 0.86 \begin {gather*} {\left (x^{2} + 2\right )} e^{\left (e^{x}\right )} - e^{\left (e^{x}\right )} \log \left (4 \, e^{\left (x^{2}\right )} + e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{x-x^2}\,\left (2\,x-1\right )+{\mathrm {e}}^x\,\left (2\,{\mathrm {e}}^{x-x^2}+8\right )\right )-\ln \left ({\mathrm {e}}^{x-x^2}+4\right )\,{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{x-x^2}+4\right )}{{\mathrm {e}}^{x-x^2}+4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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