Optimal. Leaf size=17 \[ 16-x+\frac {e^{e^x} x}{-3+x} \]
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Rubi [F]
time = 0.22, 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 {-9+6 x-x^2+e^{e^x} \left (-3+e^x \left (-3 x+x^2\right )\right )}{9-6 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9+6 x-x^2+e^{e^x} \left (-3+e^x \left (-3 x+x^2\right )\right )}{(-3+x)^2} \, dx\\ &=\int \left (-\frac {9}{(-3+x)^2}-\frac {3 e^{e^x}}{(-3+x)^2}+\frac {6 x}{(-3+x)^2}+\frac {e^{e^x+x} x}{-3+x}-\frac {x^2}{(-3+x)^2}\right ) \, dx\\ &=-\frac {9}{3-x}-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+6 \int \frac {x}{(-3+x)^2} \, dx+\int \frac {e^{e^x+x} x}{-3+x} \, dx-\int \frac {x^2}{(-3+x)^2} \, dx\\ &=-\frac {9}{3-x}-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+6 \int \left (\frac {3}{(-3+x)^2}+\frac {1}{-3+x}\right ) \, dx-\int \left (1+\frac {9}{(-3+x)^2}+\frac {6}{-3+x}\right ) \, dx+\int \left (e^{e^x+x}+\frac {3 e^{e^x+x}}{-3+x}\right ) \, dx\\ &=-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx+\int e^{e^x+x} \, dx\\ &=-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx+\text {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}-x-3 \int \frac {e^{e^x}}{(-3+x)^2} \, dx+3 \int \frac {e^{e^x+x}}{-3+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x \left (-3-e^{e^x}+x\right )}{-3+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 15, normalized size = 0.88
method | result | size |
risch | \(-x +\frac {{\mathrm e}^{{\mathrm e}^{x}} x}{x -3}\) | \(15\) |
norman | \(\frac {x \,{\mathrm e}^{{\mathrm e}^{x}}-x^{2}+9}{x -3}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 14, normalized size = 0.82 \begin {gather*} -x + \frac {x e^{\left (e^{x}\right )}}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 20, normalized size = 1.18 \begin {gather*} -\frac {x^{2} - x e^{\left (e^{x}\right )} - 3 \, x}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 10, normalized size = 0.59 \begin {gather*} - x + \frac {x e^{e^{x}}}{x - 3} \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. 33 vs.
\(2 (15) = 30\).
time = 0.40, size = 33, normalized size = 1.94 \begin {gather*} -\frac {x^{2} e^{x} - x e^{\left (x + e^{x}\right )} - 3 \, x e^{x}}{x e^{x} - 3 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 15, normalized size = 0.88 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^{{\mathrm {e}}^x}-x+3\right )}{x-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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