Optimal. Leaf size=22 \[ e^x-\frac {x^4}{\log ^2\left (-3+e^{1+x}-x\right )} \]
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Rubi [F] time = 2.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 x^4+2 e^{1+x} x^4+\left (12 x^3-4 e^{1+x} x^3+4 x^4\right ) \log \left (-3+e^{1+x}-x\right )+\left (e^{1+2 x}+e^x (-3-x)\right ) \log ^3\left (-3+e^{1+x}-x\right )}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {2 \left (-1+e^{1+x}\right ) x^4}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )}-\frac {4 x^3}{\log ^2\left (-3+e^{1+x}-x\right )}\right ) \, dx\\ &=2 \int \frac {\left (-1+e^{1+x}\right ) x^4}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )} \, dx-4 \int \frac {x^3}{\log ^2\left (-3+e^{1+x}-x\right )} \, dx+\int e^x \, dx\\ &=e^x+2 \int \left (\frac {x^4}{\log ^3\left (-3+e^{1+x}-x\right )}+\frac {x^4 (2+x)}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )}\right ) \, dx-4 \int \frac {x^3}{\log ^2\left (-3+e^{1+x}-x\right )} \, dx\\ &=e^x+2 \int \frac {x^4}{\log ^3\left (-3+e^{1+x}-x\right )} \, dx+2 \int \frac {x^4 (2+x)}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )} \, dx-4 \int \frac {x^3}{\log ^2\left (-3+e^{1+x}-x\right )} \, dx\\ &=e^x+2 \int \left (\frac {2 x^4}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )}+\frac {x^5}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )}\right ) \, dx+2 \int \frac {x^4}{\log ^3\left (-3+e^{1+x}-x\right )} \, dx-4 \int \frac {x^3}{\log ^2\left (-3+e^{1+x}-x\right )} \, dx\\ &=e^x+2 \int \frac {x^4}{\log ^3\left (-3+e^{1+x}-x\right )} \, dx+2 \int \frac {x^5}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )} \, dx+4 \int \frac {x^4}{\left (-3+e^{1+x}-x\right ) \log ^3\left (-3+e^{1+x}-x\right )} \, dx-4 \int \frac {x^3}{\log ^2\left (-3+e^{1+x}-x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.33, size = 22, normalized size = 1.00 \begin {gather*} e^x-\frac {x^4}{\log ^2\left (-3+e^{1+x}-x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 41, normalized size = 1.86 \begin {gather*} -\frac {{\left (x^{4} e - e^{\left (x + 1\right )} \log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}\right )} e^{\left (-1\right )}}{\log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 34, normalized size = 1.55 \begin {gather*} -\frac {x^{4} - e^{x} \log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}}{\log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 21, normalized size = 0.95
method | result | size |
risch | \({\mathrm e}^{x}-\frac {x^{4}}{\ln \left ({\mathrm e}^{x +1}-3-x \right )^{2}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 34, normalized size = 1.55 \begin {gather*} -\frac {x^{4} - e^{x} \log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}}{\log \left (-x + e^{\left (x + 1\right )} - 3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 274, normalized size = 12.45 \begin {gather*} {\mathrm {e}}^x-\frac {x^4+\frac {2\,x^3\,\ln \left (\mathrm {e}\,{\mathrm {e}}^x-x-3\right )\,\left (x-{\mathrm {e}}^{x+1}+3\right )}{{\mathrm {e}}^{x+1}-1}}{{\ln \left (\mathrm {e}\,{\mathrm {e}}^x-x-3\right )}^2}+\frac {\frac {2\,x^3\,\left (x-{\mathrm {e}}^{x+1}+3\right )}{{\mathrm {e}}^{x+1}-1}-\frac {2\,x^2\,\ln \left (\mathrm {e}\,{\mathrm {e}}^x-x-3\right )\,\left (x-{\mathrm {e}}^{x+1}+3\right )\,\left (4\,x-12\,{\mathrm {e}}^{x+1}+3\,{\mathrm {e}}^{2\,x+2}-2\,x\,{\mathrm {e}}^{x+1}+x^2\,{\mathrm {e}}^{x+1}+9\right )}{{\left ({\mathrm {e}}^{x+1}-1\right )}^3}}{\ln \left (\mathrm {e}\,{\mathrm {e}}^x-x-3\right )}-6\,x^2-\frac {2\,{\mathrm {e}}^{-1}\,\left (-x^4+5\,x^3+12\,x^2\right )}{{\mathrm {e}}^{-1}-{\mathrm {e}}^x}-\frac {2\,{\mathrm {e}}^{-2}\,\left (-x^5+x^4+12\,x^3+12\,x^2\right )}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{x-1}+{\mathrm {e}}^{-2}}+\frac {2\,{\mathrm {e}}^{-3}\,\left (x^5+4\,x^4+4\,x^3\right )}{{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^{x-2}-{\mathrm {e}}^{-3}-3\,{\mathrm {e}}^{2\,x-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 19, normalized size = 0.86 \begin {gather*} - \frac {x^{4}}{\log {\left (- x + e e^{x} - 3 \right )}^{2}} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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