Optimal. Leaf size=26 \[ -\frac {e+x}{x}+e^{-10-x} \left (2+\frac {x^2}{e^5}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps
used = 8, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6874, 2225,
2207} \begin {gather*} e^{-x-15} x^2+2 e^{-x-10}-\frac {e}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2 e^{-10-x}+\frac {e}{x^2}+2 e^{-15-x} x-e^{-15-x} x^2\right ) \, dx\\ &=-\frac {e}{x}-2 \int e^{-10-x} \, dx+2 \int e^{-15-x} x \, dx-\int e^{-15-x} x^2 \, dx\\ &=2 e^{-10-x}-\frac {e}{x}-2 e^{-15-x} x+e^{-15-x} x^2+2 \int e^{-15-x} \, dx-2 \int e^{-15-x} x \, dx\\ &=-2 e^{-15-x}+2 e^{-10-x}-\frac {e}{x}+e^{-15-x} x^2-2 \int e^{-15-x} \, dx\\ &=2 e^{-10-x}-\frac {e}{x}+e^{-15-x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 27, normalized size = 1.04 \begin {gather*} 2 e^{-10-x}-\frac {e}{x}+e^{-15-x} x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.29, size = 128, normalized size = 4.92
method | result | size |
risch | \(-\frac {{\mathrm e}}{x}+\left (2 \,{\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{-x -15}\) | \(24\) |
norman | \(\frac {\left ({\mathrm e}^{-5} x^{3}+2 x -{\mathrm e} \,{\mathrm e}^{x +10}\right ) {\mathrm e}^{-x -10}}{x}\) | \(31\) |
derivativedivides | \({\mathrm e}^{-5} \left (-\frac {{\mathrm e} \,{\mathrm e}^{5}}{x}+660 \,{\mathrm e}^{-x -10}-42 \left (x +31\right ) {\mathrm e}^{-x -10}+\left (\left (x +10\right )^{2}+22 x +542\right ) {\mathrm e}^{-x -10}-200 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{-x -10}}{x}+{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )+40 \,{\mathrm e}^{5} \left (-\frac {10 \,{\mathrm e}^{-x -10}}{x}+9 \,{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )-2 \,{\mathrm e}^{5} \left (-{\mathrm e}^{-x -10}-\frac {100 \,{\mathrm e}^{-x -10}}{x}+80 \,{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )\right )\) | \(128\) |
default | \({\mathrm e}^{-5} \left (-\frac {{\mathrm e} \,{\mathrm e}^{5}}{x}+660 \,{\mathrm e}^{-x -10}-42 \left (x +31\right ) {\mathrm e}^{-x -10}+\left (\left (x +10\right )^{2}+22 x +542\right ) {\mathrm e}^{-x -10}-200 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{-x -10}}{x}+{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )+40 \,{\mathrm e}^{5} \left (-\frac {10 \,{\mathrm e}^{-x -10}}{x}+9 \,{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )-2 \,{\mathrm e}^{5} \left (-{\mathrm e}^{-x -10}-\frac {100 \,{\mathrm e}^{-x -10}}{x}+80 \,{\mathrm e}^{-10} \expIntegral \left (1, x\right )\right )\right )\) | \(128\) |
meijerg | \({\mathrm e}^{-x -9+x \,{\mathrm e}^{-10}} \left (-{\mathrm e}^{10}+1\right ) \left (\frac {{\mathrm e}^{10} \left (2-2 x \,{\mathrm e}^{-10} \left (-{\mathrm e}^{10}+1\right )\right )}{2 x \left (-{\mathrm e}^{10}+1\right )}-\frac {{\mathrm e}^{10-x \,{\mathrm e}^{-10} \left (-{\mathrm e}^{10}+1\right )}}{x \left (-{\mathrm e}^{10}+1\right )}+\ln \left (x \,{\mathrm e}^{-10} \left (-{\mathrm e}^{10}+1\right )\right )+\expIntegral \left (1, x \,{\mathrm e}^{-10} \left (-{\mathrm e}^{10}+1\right )\right )+11-\ln \left (x \right )-\ln \left (-{\mathrm e}^{10}+1\right )-\frac {{\mathrm e}^{10}}{x \left (-{\mathrm e}^{10}+1\right )}\right )-2 \,{\mathrm e}^{-x +x \,{\mathrm e}^{-10}} \left (1-{\mathrm e}^{-x \,{\mathrm e}^{-10}}\right )-{\mathrm e}^{15-x +x \,{\mathrm e}^{-10}} \left (2-\frac {\left (3 x^{2} {\mathrm e}^{-20}+6 x \,{\mathrm e}^{-10}+6\right ) {\mathrm e}^{-x \,{\mathrm e}^{-10}}}{3}\right )+2 \,{\mathrm e}^{-x +5+x \,{\mathrm e}^{-10}} \left (1-\frac {\left (2+2 x \,{\mathrm e}^{-10}\right ) {\mathrm e}^{-x \,{\mathrm e}^{-10}}}{2}\right )\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 42, normalized size = 1.62 \begin {gather*} {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x - 15\right )} - 2 \, {\left (x + 1\right )} e^{\left (-x - 15\right )} - \frac {e}{x} + 2 \, e^{\left (-x - 10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 28, normalized size = 1.08 \begin {gather*} \frac {{\left (x^{3} e + 2 \, x e^{6} - e^{\left (x + 17\right )}\right )} e^{\left (-x - 16\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.85 \begin {gather*} \frac {\left (x^{2} + 2 e^{5}\right ) e^{- x - 10}}{e^{5}} - \frac {e}{x} \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. 72 vs.
\(2 (25) = 50\).
time = 0.42, size = 72, normalized size = 2.77 \begin {gather*} \frac {{\left (x + 16\right )}^{3} e^{\left (-x - 15\right )} - 48 \, {\left (x + 16\right )}^{2} e^{\left (-x - 15\right )} + 2 \, {\left (x + 16\right )} e^{\left (-x - 10\right )} + 768 \, {\left (x + 16\right )} e^{\left (-x - 15\right )} - e - 32 \, e^{\left (-x - 10\right )} - 4096 \, e^{\left (-x - 15\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 26, normalized size = 1.00 \begin {gather*} 2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-10}-\frac {\mathrm {e}}{x}+x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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