Optimal. Leaf size=29 \[ \frac {e^{-x} \left (9-e^{\frac {1}{5} \left (x-\left (1+e^5\right ) x\right )}\right )}{x} \]
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Rubi [A]
time = 0.23, antiderivative size = 28, normalized size of antiderivative = 0.97, number of steps
used = 5, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {12, 6874, 2228}
\begin {gather*} \frac {9 e^{-x}}{x}-\frac {e^{-\frac {1}{5} \left (5+e^5\right ) x}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2228
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {e^{-x} \left (-45-45 x+e^{-\frac {e^5 x}{5}} \left (5+5 x+e^5 x\right )\right )}{x^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {45 e^{-x} (1+x)}{x^2}+\frac {e^{-x-\frac {e^5 x}{5}} \left (5+\left (5+e^5\right ) x\right )}{x^2}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{-x-\frac {e^5 x}{5}} \left (5+\left (5+e^5\right ) x\right )}{x^2} \, dx-9 \int \frac {e^{-x} (1+x)}{x^2} \, dx\\ &=\frac {9 e^{-x}}{x}-\frac {e^{-\frac {1}{5} \left (5+e^5\right ) x}}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.30, size = 23, normalized size = 0.79 \begin {gather*} \frac {e^{-x} \left (9-e^{-\frac {e^5 x}{5}}\right )}{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 = 0.09, size = 82, normalized size = 2.83
method | result | size |
norman | \(\frac {\left (9-{\mathrm e}^{-\frac {x \,{\mathrm e}^{5}}{5}}\right ) {\mathrm e}^{-x}}{x}\) | \(19\) |
risch | \(\frac {9 \,{\mathrm e}^{-x}}{x}-\frac {{\mathrm e}^{-\frac {x \left ({\mathrm e}^{5}+5\right )}{5}}}{x}\) | \(24\) |
default | \(-\frac {{\mathrm e}^{5} \expIntegral \left (1, -\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) x \right )}{5}+\frac {9 \,{\mathrm e}^{-x}}{x}+\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) \left (-\frac {{\mathrm e}^{\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) x}}{\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) x}-\expIntegral \left (1, -\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) x \right )\right )-\expIntegral \left (1, -\left (-1-\frac {{\mathrm e}^{5}}{5}\right ) x \right )\) | \(82\) |
meijerg | \(-\frac {9 \left (-2 x +2\right )}{2 x}+\frac {9 \,{\mathrm e}^{-x}}{x}-4+\frac {9}{x}+\frac {{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right ) \left (\frac {5 \,{\mathrm e}^{-5} \left (2-\frac {2 x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )}{2 x \left (1+5 \,{\mathrm e}^{-5}\right )}-\frac {5 \,{\mathrm e}^{-5-\frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}}}{x \left (1+5 \,{\mathrm e}^{-5}\right )}+\ln \left (\frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )+\expIntegral \left (1, \frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )-4-\ln \left (x \right )+\ln \left (5\right )-\ln \left (1+5 \,{\mathrm e}^{-5}\right )-\frac {5 \,{\mathrm e}^{-5}}{x \left (1+5 \,{\mathrm e}^{-5}\right )}\right )}{5}-\ln \left (\frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )-\expIntegral \left (1, \frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )+\ln \left (x \right )-\ln \left (5\right )+\ln \left (1+5 \,{\mathrm e}^{-5}\right )+\frac {{\mathrm e}^{5} \left (-\ln \left (\frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )-\expIntegral \left (1, \frac {x \,{\mathrm e}^{5} \left (1+5 \,{\mathrm e}^{-5}\right )}{5}\right )+\ln \left (x \right )-\ln \left (5\right )+5+\ln \left (1+5 \,{\mathrm e}^{-5}\right )\right )}{5}\) | \(239\) |
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 = 0.52, size = 47, normalized size = 1.62 \begin {gather*} \frac {1}{5} \, {\rm Ei}\left (-\frac {1}{5} \, x {\left (e^{5} + 5\right )}\right ) e^{5} - \frac {1}{5} \, {\left (e^{5} + 5\right )} \Gamma \left (-1, \frac {1}{5} \, x {\left (e^{5} + 5\right )}\right ) + {\rm Ei}\left (-\frac {1}{5} \, x {\left (e^{5} + 5\right )}\right ) - 9 \, {\rm Ei}\left (-x\right ) + 9 \, \Gamma \left (-1, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 17, normalized size = 0.59 \begin {gather*} -\frac {{\left (e^{\left (-\frac {1}{5} \, x e^{5}\right )} - 9\right )} e^{\left (-x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 20, normalized size = 0.69 \begin {gather*} \frac {9 e^{- x}}{x} - \frac {e^{- x} e^{- \frac {x e^{5}}{5}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 22, normalized size = 0.76 \begin {gather*} -\frac {e^{\left (-\frac {1}{5} \, x e^{5} - x\right )} - 9 \, e^{\left (-x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 23, normalized size = 0.79 \begin {gather*} \frac {9\,{\mathrm {e}}^{-x}-{\mathrm {e}}^{-x-\frac {x\,{\mathrm {e}}^5}{5}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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