Optimal. Leaf size=26 \[ 1+e^x \left (x-2 \left (1+\left (e^{5/x}-x\right )^2+x\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(26)=52\).
time = 0.38, antiderivative size = 61, normalized size of antiderivative = 2.35, number of steps
used = 10, number of rules used = 5, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6874, 2225,
2207, 6838, 2326} \begin {gather*} -2 e^x x^2-\frac {4 e^{x+\frac {5}{x}} \left (5-x^2\right )}{\left (1-\frac {5}{x^2}\right ) x}-e^x x-2 e^x-2 e^{x+\frac {10}{x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2326
Rule 6838
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3 e^x-5 e^x x-2 e^x x^2-\frac {2 e^{\frac {10}{x}+x} \left (-10+x^2\right )}{x^2}+\frac {4 e^{\frac {5}{x}+x} \left (-5+x+x^2\right )}{x}\right ) \, dx\\ &=-\left (2 \int e^x x^2 \, dx\right )-2 \int \frac {e^{\frac {10}{x}+x} \left (-10+x^2\right )}{x^2} \, dx-3 \int e^x \, dx+4 \int \frac {e^{\frac {5}{x}+x} \left (-5+x+x^2\right )}{x} \, dx-5 \int e^x x \, dx\\ &=-3 e^x-2 e^{\frac {10}{x}+x}-5 e^x x-2 e^x x^2-\frac {4 e^{\frac {5}{x}+x} \left (5-x^2\right )}{\left (1-\frac {5}{x^2}\right ) x}+4 \int e^x x \, dx+5 \int e^x \, dx\\ &=2 e^x-2 e^{\frac {10}{x}+x}-e^x x-2 e^x x^2-\frac {4 e^{\frac {5}{x}+x} \left (5-x^2\right )}{\left (1-\frac {5}{x^2}\right ) x}-4 \int e^x \, dx\\ &=-2 e^x-2 e^{\frac {10}{x}+x}-e^x x-2 e^x x^2-\frac {4 e^{\frac {5}{x}+x} \left (5-x^2\right )}{\left (1-\frac {5}{x^2}\right ) x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 32, normalized size = 1.23 \begin {gather*} -e^x \left (2+2 e^{10/x}+x-4 e^{5/x} x+2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 40, normalized size = 1.54
method | result | size |
risch | \(\left (-2 x^{2}-x -2\right ) {\mathrm e}^{x}-2 \,{\mathrm e}^{\frac {x^{2}+10}{x}}+4 x \,{\mathrm e}^{\frac {x^{2}+5}{x}}\) | \(40\) |
norman | \(\frac {-2 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x^{3}-2 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {10}{x}}+4 \,{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {5}{x}}}{x}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 45, normalized size = 1.73 \begin {gather*} 4 \, x e^{\left (x + \frac {5}{x}\right )} - 2 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 5 \, {\left (x - 1\right )} e^{x} - 2 \, e^{\left (x + \frac {10}{x}\right )} - 3 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 29, normalized size = 1.12 \begin {gather*} -{\left (2 \, x^{2} - 4 \, x e^{\frac {5}{x}} + x + 2 \, e^{\frac {10}{x}} + 2\right )} e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.51, size = 26, normalized size = 1.00 \begin {gather*} \left (- 2 x^{2} + 4 x e^{\frac {5}{x}} - x - 2 e^{\frac {10}{x}} - 2\right ) e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 42, normalized size = 1.62 \begin {gather*} -2 \, x^{2} e^{x} - x e^{x} + 4 \, x e^{\left (\frac {x^{2} + 5}{x}\right )} - 2 \, e^{x} - 2 \, e^{\left (\frac {x^{2} + 10}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 29, normalized size = 1.12 \begin {gather*} -{\mathrm {e}}^x\,\left (x+2\,{\mathrm {e}}^{10/x}-4\,x\,{\mathrm {e}}^{5/x}+2\,x^2+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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