Optimal. Leaf size=29 \[ \left (-3-e^{-2+x}+4 e^{-e^4 (x+\log (2))}+\frac {5}{x}\right ) x \]
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Rubi [A]
time = 0.27, antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps
used = 11, number of rules used = 5, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2306, 12,
6874, 2225, 2207} \begin {gather*} -e^{x-2} x+2^{2-e^4} e^{-e^4 x} x-3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2225
Rule 2306
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int 2^{-e^4} e^{-e^4 x} \left (4+e^{e^4 x+e^4 \log (2)} \left (-3+e^{-2+x} (-1-x)\right )-4 e^4 x\right ) \, dx\\ &=2^{-e^4} \int e^{-e^4 x} \left (4+e^{e^4 x+e^4 \log (2)} \left (-3+e^{-2+x} (-1-x)\right )-4 e^4 x\right ) \, dx\\ &=2^{-e^4} \int \left (4 e^{-e^4 x}-4 e^{4-e^4 x} x-\frac {2^{e^4} \left (3 e^2+e^x+e^x x\right )}{e^2}\right ) \, dx\\ &=2^{2-e^4} \int e^{-e^4 x} \, dx-2^{2-e^4} \int e^{4-e^4 x} x \, dx-\frac {\int \left (3 e^2+e^x+e^x x\right ) \, dx}{e^2}\\ &=-2^{2-e^4} e^{-4-e^4 x}-3 x+2^{2-e^4} e^{-e^4 x} x-\frac {2^{2-e^4} \int e^{4-e^4 x} \, dx}{e^4}-\frac {\int e^x \, dx}{e^2}-\frac {\int e^x x \, dx}{e^2}\\ &=-e^{-2+x}-3 x-e^{-2+x} x+2^{2-e^4} e^{-e^4 x} x+\frac {\int e^x \, dx}{e^2}\\ &=-3 x-e^{-2+x} x+2^{2-e^4} e^{-e^4 x} x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.65, size = 41, normalized size = 1.41 \begin {gather*} 2^{-e^4} \left (-3 2^{e^4} x-2^{e^4} e^{-2+x} x+4 e^{-e^4 x} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs.
\(2(27)=54\).
time = 0.05, size = 77, normalized size = 2.66
method | result | size |
risch | \(-x \,{\mathrm e}^{x -2}-3 x +4 x 2^{-{\mathrm e}^{4}} {\mathrm e}^{-x \,{\mathrm e}^{4}}\) | \(27\) |
norman | \(\left (4 x -3 x \,{\mathrm e}^{{\mathrm e}^{4} \ln \left (2\right )+x \,{\mathrm e}^{4}}-{\mathrm e}^{x -2} {\mathrm e}^{{\mathrm e}^{4} \ln \left (2\right )+x \,{\mathrm e}^{4}} x \right ) {\mathrm e}^{-{\mathrm e}^{4} \ln \left (2\right )-x \,{\mathrm e}^{4}}\) | \(51\) |
default | \(-3 x -4 \,2^{-{\mathrm e}^{4}} {\mathrm e}^{-4} {\mathrm e}^{-x \,{\mathrm e}^{4}}-{\mathrm e}^{x} {\mathrm e}^{-2}-{\mathrm e}^{-2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-4 \,2^{-{\mathrm e}^{4}} {\mathrm e}^{-4} \left (-x \,{\mathrm e}^{4} {\mathrm e}^{-x \,{\mathrm e}^{4}}-{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (26) = 52\).
time = 0.53, size = 58, normalized size = 2.00 \begin {gather*} {\left (x e^{4} + 1\right )} 2^{-e^{4} + 2} e^{\left (-x e^{4} - 4\right )} - {\left (x - 1\right )} e^{\left (x - 2\right )} - 3 \, x - 4 \, e^{\left (-x e^{4} - e^{4} \log \left (2\right ) - 4\right )} - e^{\left (x - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 41, normalized size = 1.41 \begin {gather*} -{\left ({\left (x e^{\left (x - 2\right )} + 3 \, x\right )} e^{\left (x e^{4} + e^{4} \log \left (2\right )\right )} - 4 \, x\right )} e^{\left (-x e^{4} - e^{4} \log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.28, size = 36, normalized size = 1.24 \begin {gather*} \frac {- \frac {2^{e^{4}} x e^{x}}{e^{2}} - 3 \cdot 2^{e^{4}} x + 4 x e^{- x e^{4}}}{2^{e^{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (26) = 52\).
time = 0.39, size = 73, normalized size = 2.52 \begin {gather*} -{\left (3 \, {\left (x - 2\right )} e^{8} - 4 \, {\left (x - 2\right )} e^{\left (-{\left (x - 2\right )} e^{4} - e^{4} \log \left (2\right ) - 2 \, e^{4} + 8\right )} + {\left (x - 2\right )} e^{\left (x + 6\right )} - 8 \, e^{\left (-{\left (x - 2\right )} e^{4} - e^{4} \log \left (2\right ) - 2 \, e^{4} + 8\right )} + 2 \, e^{\left (x + 6\right )}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 23, normalized size = 0.79 \begin {gather*} -x\,\left ({\mathrm {e}}^{x-2}-\frac {4\,{\mathrm {e}}^{-x\,{\mathrm {e}}^4}}{2^{{\mathrm {e}}^4}}+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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