Optimal. Leaf size=26 \[ 2-e^{\left (-4+5 e^5 (4-x)-\frac {5}{x}\right ) x}+x \]
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Rubi [A]
time = 0.05, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2276, 2268}
\begin {gather*} x-e^{-5 e^5 x^2-4 \left (1-5 e^5\right ) x-5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2268
Rule 2276
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x+\int e^{-5-4 x+e^5 \left (20 x-5 x^2\right )} \left (4+e^5 (-20+10 x)\right ) \, dx\\ &=x+\int e^{-5-4 \left (1-5 e^5\right ) x-5 e^5 x^2} \left (4 \left (1-5 e^5\right )+10 e^5 x\right ) \, dx\\ &=-e^{-5-4 \left (1-5 e^5\right ) x-5 e^5 x^2}+x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.96 \begin {gather*} -e^{-5-4 x+20 e^5 x-5 e^5 x^2}+x \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 = 0.13, size = 191, normalized size = 7.35
method | result | size |
norman | \(x -{\mathrm e}^{\left (-5 x^{2}+20 x \right ) {\mathrm e}^{5}-4 x -5}\) | \(23\) |
risch | \(x -{\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+20 x \,{\mathrm e}^{5}-4 x -5}\) | \(23\) |
default | \(x +\frac {2 \sqrt {\pi }\, {\mathrm e}^{-5+\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{5}-2 \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )-{\mathrm e}^{-5} {\mathrm e}^{-5 x^{2} {\mathrm e}^{5}+\left (20 \,{\mathrm e}^{5}-4\right ) x}+\frac {\left (20 \,{\mathrm e}^{5}-4\right ) {\mathrm e}^{-5} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (20 \,{\mathrm e}^{5}-4\right )^{2} {\mathrm e}^{-5}}{20}} \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}} \erf \left (\sqrt {5}\, {\mathrm e}^{\frac {5}{2}} x -\frac {\left (20 \,{\mathrm e}^{5}-4\right ) \sqrt {5}\, {\mathrm e}^{-\frac {5}{2}}}{10}\right )}{10}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.54, size = 223, normalized size = 8.58 \begin {gather*} -2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {5}{2}\right )} + \frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (\sqrt {5} x e^{\frac {5}{2}} - \frac {2}{5} \, \sqrt {5} {\left (5 \, e^{5} - 1\right )} e^{\left (-\frac {5}{2}\right )}\right ) e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )} - \frac {15}{2}\right )} + \frac {\sqrt {5} {\left (\frac {2 \, \sqrt {5} \sqrt {\frac {1}{5}} \sqrt {\pi } {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{5}} \sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} e^{\left (-\frac {5}{2}\right )}\right ) - 1\right )} {\left (5 \, e^{5} - 1\right )} e^{\frac {5}{2}}}{\sqrt {{\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2}} \left (-e^{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {5} e^{\left (-\frac {1}{5} \, {\left (5 \, x e^{5} - 10 \, e^{5} + 2\right )}^{2} e^{\left (-5\right )} + 5\right )}}{\left (-e^{5}\right )^{\frac {3}{2}}}\right )} e^{\left (\frac {4}{5} \, {\left (5 \, e^{5} - 1\right )}^{2} e^{\left (-5\right )}\right )}}{5 \, \sqrt {-e^{5}}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 21, normalized size = 0.81 \begin {gather*} x - e^{\left (-5 \, {\left (x^{2} - 4 \, x\right )} e^{5} - 4 \, x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.73 \begin {gather*} x - e^{- 4 x + \left (- 5 x^{2} + 20 x\right ) e^{5} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.42, size = 120, normalized size = 4.62 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {1}{5} \, {\left (100 \, e^{10} - 65 \, e^{5} + 4\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} - \frac {1}{5} \, {\left (2 \, \sqrt {5} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{5} \, \sqrt {5} {\left (2 \, {\left (5 \, e^{5} - 1\right )} e^{\left (-5\right )} - 5 \, x\right )} e^{\frac {5}{2}}\right ) e^{\left (\frac {4}{5} \, {\left (25 \, e^{10} - 10 \, e^{5} + 1\right )} e^{\left (-5\right )} - \frac {5}{2}\right )} + 5 \, e^{\left (-5 \, x^{2} e^{5} + 20 \, x e^{5} - 4 \, x\right )}\right )} e^{\left (-5\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.30, size = 24, normalized size = 0.92 \begin {gather*} x-{\mathrm {e}}^{-5\,x^2\,{\mathrm {e}}^5}\,{\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{20\,x\,{\mathrm {e}}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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