Optimal. Leaf size=24 \[ \frac {e^{-3+x} x}{i \pi +x+\log (3)-5 \log ^2(3)} \]
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Rubi [A]
time = 0.28, antiderivative size = 45, normalized size of antiderivative = 1.88, number of steps
used = 9, number of rules used = 7, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6, 1600, 27,
2230, 2225, 2208, 2209} \begin {gather*} e^{x-3}-\frac {e^{x-3} (\log (3) (1-\log (243))+i \pi )}{x+i \pi -5 \log ^2(3)+\log (3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 27
Rule 1600
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-3+x} x \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{x^3+2 x^2 (i \pi +\log (3))+\log ^2(3) \left (-10 x^2-10 x (i \pi +\log (3))\right )+x \left (25 \log ^4(3)+(i \pi +\log (3))^2\right )} \, dx\\ &=\int \frac {e^{-3+x} \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{-\pi ^2+x^2+2 i \pi \log (3)+\log ^2(3)-10 i \pi \log ^2(3)-10 \log ^3(3)+25 \log ^4(3)+x \left (2 i \pi +2 \log (3)-10 \log ^2(3)\right )} \, dx\\ &=\int \frac {e^{-3+x} \left (x^2+(-5-5 x) \log ^2(3)+(1+x) (i \pi +\log (3))\right )}{\left (x+i \left (\pi -i \log (3)+5 i \log ^2(3)\right )\right )^2} \, dx\\ &=\int \left (e^{-3+x}+\frac {i e^{-3+x} (\pi -i \log (3) (1-\log (243)))}{\left (i \pi +x+\log (3)-5 \log ^2(3)\right )^2}+\frac {e^{-3+x} (-i \pi -\log (3)+\log (3) \log (243))}{i \pi +x+\log (3)-5 \log ^2(3)}\right ) \, dx\\ &=(-i \pi -\log (3) (1-\log (243))) \int \frac {e^{-3+x}}{i \pi +x+\log (3)-5 \log ^2(3)} \, dx+(i \pi +\log (3)-\log (3) \log (243)) \int \frac {e^{-3+x}}{\left (i \pi +x+\log (3)-5 \log ^2(3)\right )^2} \, dx+\int e^{-3+x} \, dx\\ &=e^{-3+x}+\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3) (1-\log (243)))-\frac {e^{-3+x} (i \pi +\log (3) (1-\log (243)))}{i \pi +x+\log (3)-5 \log ^2(3)}+(i \pi +\log (3)-\log (3) \log (243)) \int \frac {e^{-3+x}}{i \pi +x+\log (3)-5 \log ^2(3)} \, dx\\ &=e^{-3+x}+\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3) (1-\log (243)))-\frac {e^{-3+x} (i \pi +\log (3) (1-\log (243)))}{i \pi +x+\log (3)-5 \log ^2(3)}-\frac {1}{3} e^{-3+5 \log ^2(3)} \text {Ei}\left (i \pi +x+\log (3)-5 \log ^2(3)\right ) (i \pi +\log (3)-\log (3) \log (243))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 24, normalized size = 1.00 \begin {gather*} \frac {e^{-3+x} x}{i \pi +x+\log (3)-5 \log ^2(3)} \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 = 7.81, size = 1397, normalized size = 58.21
method | result | size |
risch | \(-\frac {i x \,{\mathrm e}^{x -3}}{\pi +5 i \ln \left (3\right )^{2}-i \ln \left (3\right )-i x}\) | \(29\) |
norman | \(\frac {x \,{\mathrm e}^{\ln \left (x \right )-3+x}+\left (-i \pi -5 \ln \left (3\right )^{2}+\ln \left (3\right )\right ) {\mathrm e}^{\ln \left (x \right )-3+x}}{25 \ln \left (3\right )^{4}-10 \ln \left (3\right )^{3}-10 x \ln \left (3\right )^{2}+\pi ^{2}+\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}}\) | \(68\) |
default | \(\text {Expression too large to display}\) | \(1397\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 26, normalized size = 1.08 \begin {gather*} \frac {x e^{x}}{{\left (i \, \pi - 5 \, \log \left (3\right )^{2} + \log \left (3\right )\right )} e^{3} + x e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 1.12 \begin {gather*} -\frac {e^{\left (x + \log \left (x\right ) - 3\right )}}{-i \, \pi + 5 \, \log \left (3\right )^{2} - x - \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 32, normalized size = 1.33 \begin {gather*} \frac {x e^{x}}{x e^{3} - 5 e^{3} \log {\left (3 \right )}^{2} + e^{3} \log {\left (3 \right )} + i \pi e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 31, normalized size = 1.29 \begin {gather*} -\frac {i \, x e^{x}}{5 i \, e^{3} \log \left (3\right )^{2} + \pi e^{3} - i \, x e^{3} - i \, e^{3} \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{x+\ln \left (x\right )-3}\,\left (x^2+\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\,\left (x+1\right )-{\ln \left (3\right )}^2\,\left (5\,x+5\right )\right )}{25\,x\,{\ln \left (3\right )}^4-{\ln \left (3\right )}^2\,\left (10\,x\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+10\,x^2\right )+x\,{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}^2+2\,x^2\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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