Optimal. Leaf size=25 \[ 3+2 x+\frac {e^{-4+5 x} (-2+3 (3+x) \log (2))}{x} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.16, antiderivative size = 63, normalized size of antiderivative = 2.52, number of steps
used = 8, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {14, 2230, 2208,
2209, 2225} \begin {gather*} -\frac {5 (2-\log (512)) \text {ExpIntegralEi}(5 x)}{e^4}+\frac {5 (2-9 \log (2)) \text {ExpIntegralEi}(5 x)}{e^4}+2 x+3 e^{5 x-4} \log (2)-\frac {e^{5 x-4} (2-\log (512))}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {e^{-4+5 x} \left (2-9 \log (2)+15 x^2 \log (2)-5 x (2-\log (512))\right )}{x^2}\right ) \, dx\\ &=2 x+\int \frac {e^{-4+5 x} \left (2-9 \log (2)+15 x^2 \log (2)-5 x (2-\log (512))\right )}{x^2} \, dx\\ &=2 x+\int \left (\frac {e^{-4+5 x} (2-9 \log (2))}{x^2}+15 e^{-4+5 x} \log (2)+\frac {5 e^{-4+5 x} (-2+\log (512))}{x}\right ) \, dx\\ &=2 x+(2-9 \log (2)) \int \frac {e^{-4+5 x}}{x^2} \, dx+(15 \log (2)) \int e^{-4+5 x} \, dx-(5 (2-\log (512))) \int \frac {e^{-4+5 x}}{x} \, dx\\ &=2 x+3 e^{-4+5 x} \log (2)-\frac {e^{-4+5 x} (2-\log (512))}{x}-\frac {5 \text {Ei}(5 x) (2-\log (512))}{e^4}+(5 (2-9 \log (2))) \int \frac {e^{-4+5 x}}{x} \, dx\\ &=2 x+\frac {5 \text {Ei}(5 x) (2-9 \log (2))}{e^4}+3 e^{-4+5 x} \log (2)-\frac {e^{-4+5 x} (2-\log (512))}{x}-\frac {5 \text {Ei}(5 x) (2-\log (512))}{e^4}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 31, normalized size = 1.24 \begin {gather*} 2 x+e^{5 x} \left (\frac {3 \log (2)}{e^4}+\frac {-2+9 \log (2)}{e^4 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 40, normalized size = 1.60
method | result | size |
risch | \(2 x +\frac {\left (3 x \ln \left (2\right )+9 \ln \left (2\right )-2\right ) {\mathrm e}^{5 x -4}}{x}\) | \(26\) |
norman | \(\frac {\left (9 \ln \left (2\right )-2\right ) {\mathrm e}^{5 x -4}+2 x^{2}+3 \ln \left (2\right ) {\mathrm e}^{5 x -4} x}{x}\) | \(35\) |
derivativedivides | \(2 x -\frac {8}{5}-\frac {2 \,{\mathrm e}^{5 x -4}}{x}+\frac {9 \ln \left (2\right ) {\mathrm e}^{5 x -4}}{x}+3 \ln \left (2\right ) {\mathrm e}^{5 x -4}\) | \(40\) |
default | \(2 x -\frac {8}{5}-\frac {2 \,{\mathrm e}^{5 x -4}}{x}+\frac {9 \ln \left (2\right ) {\mathrm e}^{5 x -4}}{x}+3 \ln \left (2\right ) {\mathrm e}^{5 x -4}\) | \(40\) |
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 = 0.29, size = 52, normalized size = 2.08 \begin {gather*} 45 \, {\rm Ei}\left (5 \, x\right ) e^{\left (-4\right )} \log \left (2\right ) - 45 \, e^{\left (-4\right )} \Gamma \left (-1, -5 \, x\right ) \log \left (2\right ) - 10 \, {\rm Ei}\left (5 \, x\right ) e^{\left (-4\right )} + 10 \, e^{\left (-4\right )} \Gamma \left (-1, -5 \, x\right ) + 3 \, e^{\left (5 \, x - 4\right )} \log \left (2\right ) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 26, normalized size = 1.04 \begin {gather*} \frac {2 \, x^{2} + {\left (3 \, {\left (x + 3\right )} \log \left (2\right ) - 2\right )} e^{\left (5 \, x - 4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 24, normalized size = 0.96 \begin {gather*} 2 x + \frac {\left (3 x \log {\left (2 \right )} - 2 + 9 \log {\left (2 \right )}\right ) e^{5 x - 4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 37, normalized size = 1.48 \begin {gather*} \frac {{\left (2 \, x^{2} e^{4} + 3 \, x e^{\left (5 \, x\right )} \log \left (2\right ) + 9 \, e^{\left (5 \, x\right )} \log \left (2\right ) - 2 \, e^{\left (5 \, x\right )}\right )} e^{\left (-4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.69, size = 27, normalized size = 1.08 \begin {gather*} 2\,x+{\mathrm {e}}^{5\,x-4}\,\ln \left (8\right )+\frac {{\mathrm {e}}^{5\,x-4}\,\left (\ln \left (512\right )-2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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