Optimal. Leaf size=25 \[ e^{-5+e^{\frac {e^{-2 x}}{3}}}-e^{(-5+x)^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {12, 6874, 2320,
2225, 2259, 2240} \begin {gather*} e^{e^{\frac {e^{-2 x}}{3}}-5}-e^{(x-5)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2225
Rule 2240
Rule 2259
Rule 2320
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{-2 x} \left (-2 e^{-5+e^{\frac {e^{-2 x}}{3}}+\frac {e^{-2 x}}{3}}+3 e^{25-8 x+x^2} (10-2 x)\right ) \, dx\\ &=\frac {1}{3} \int \left (-2 e^{-5+e^{\frac {e^{-2 x}}{3}}+\frac {e^{-2 x}}{3}-2 x}-6 e^{25-10 x+x^2} (-5+x)\right ) \, dx\\ &=-\left (\frac {2}{3} \int e^{-5+e^{\frac {e^{-2 x}}{3}}+\frac {e^{-2 x}}{3}-2 x} \, dx\right )-2 \int e^{25-10 x+x^2} (-5+x) \, dx\\ &=\frac {1}{3} \text {Subst}\left (\int e^{\frac {1}{3} \left (-15+3 e^{x/3}+x\right )} \, dx,x,e^{-2 x}\right )-2 \int e^{(-5+x)^2} (-5+x) \, dx\\ &=-e^{(-5+x)^2}+\text {Subst}\left (\int e^{-5+e^x+x} \, dx,x,\frac {e^{-2 x}}{3}\right )\\ &=-e^{(-5+x)^2}+\text {Subst}\left (\int e^{-5+x} \, dx,x,e^{\frac {e^{-2 x}}{3}}\right )\\ &=e^{-5+e^{\frac {e^{-2 x}}{3}}}-e^{(-5+x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.16, size = 25, normalized size = 1.00 \begin {gather*} e^{-5+e^{\frac {e^{-2 x}}{3}}}-e^{(-5+x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 24, normalized size = 0.96
method | result | size |
risch | \(-{\mathrm e}^{\left (x -5\right )^{2}}+{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-2 x}}{3}}-5}\) | \(20\) |
default | \(-{\mathrm e}^{x^{2}-10 x +25}+{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-2 x}}{3}}} {\mathrm e}^{-5}\) | \(24\) |
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 = 0.31, size = 59, normalized size = 2.36 \begin {gather*} -\frac {5 \, \sqrt {\pi } {\left (x - 5\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x - 5\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x - 5\right )}^{2}}} - 5 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x - 5 i\right ) - e^{\left ({\left (x - 5\right )}^{2}\right )} + e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-2 \, x\right )}\right )} - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (22) = 44\).
time = 0.35, size = 57, normalized size = 2.28 \begin {gather*} -{\left (e^{\left (x^{2} - 10 \, x + e^{\left (-2 \, x - \log \left (3\right )\right )} + 25\right )} - e^{\left (e^{\left (-2 \, x - \log \left (3\right )\right )} + e^{\left (e^{\left (-2 \, x - \log \left (3\right )\right )}\right )} - 5\right )}\right )} e^{\left (-e^{\left (-2 \, x - \log \left (3\right )\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 22, normalized size = 0.88 \begin {gather*} e^{e^{\frac {e^{- 2 x}}{3}} - 5} - e^{x^{2} - 10 x + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.27, size = 24, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-5}\,{\mathrm {e}}^{{\left ({\mathrm {e}}^{{\mathrm {e}}^{-2\,x}}\right )}^{1/3}}-{\mathrm {e}}^{-10\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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