Optimal. Leaf size=32 \[ e^{-x} \left (e^{1+x}-\frac {\log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}\right ) \]
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Rubi [A] time = 1.25, antiderivative size = 25, normalized size of antiderivative = 0.78, number of steps used = 18, number of rules used = 8, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.110, Rules used = {1593, 6688, 6742, 2178, 2177, 2197, 2554, 12} \begin {gather*} -\frac {e^{-x} \log \left (\frac {4 \log ^2(5)}{x}-\log (3)\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2177
Rule 2178
Rule 2197
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (4 \log ^2(5)+\left (\left (-x-x^2\right ) \log (3)+(4+4 x) \log ^2(5)\right ) \log \left (\frac {-x \log (3)+4 \log ^2(5)}{x}\right )\right )}{x^2 \left (-x \log (3)+4 \log ^2(5)\right )} \, dx\\ &=\int \frac {e^{-x} \left (-\frac {4 \log ^2(5)}{x \log (3)-4 \log ^2(5)}+(1+x) \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )\right )}{x^2} \, dx\\ &=\int \left (-\frac {4 e^{-x} \log ^2(5)}{x^2 \left (x \log (3)-4 \log ^2(5)\right )}+\frac {e^{-x} (1+x) \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x^2}\right ) \, dx\\ &=-\left (\left (4 \log ^2(5)\right ) \int \frac {e^{-x}}{x^2 \left (x \log (3)-4 \log ^2(5)\right )} \, dx\right )+\int \frac {e^{-x} (1+x) \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x^2} \, dx\\ &=-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}-\left (4 \log ^2(5)\right ) \int \left (-\frac {e^{-x} \log (3)}{16 x \log ^4(5)}-\frac {e^{-x}}{4 x^2 \log ^2(5)}-\frac {e^{-x} \log ^2(3)}{16 \log ^4(5) \left (-x \log (3)+4 \log ^2(5)\right )}\right ) \, dx-\int \frac {4 e^{-x} \log ^2(5)}{x^2 \left (-x \log (3)+4 \log ^2(5)\right )} \, dx\\ &=-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}+\frac {\log (3) \int \frac {e^{-x}}{x} \, dx}{4 \log ^2(5)}+\frac {\log ^2(3) \int \frac {e^{-x}}{-x \log (3)+4 \log ^2(5)} \, dx}{4 \log ^2(5)}-\left (4 \log ^2(5)\right ) \int \frac {e^{-x}}{x^2 \left (-x \log (3)+4 \log ^2(5)\right )} \, dx+\int \frac {e^{-x}}{x^2} \, dx\\ &=-\frac {e^{-x}}{x}+\frac {\text {Ei}(-x) \log (3)}{4 \log ^2(5)}-\frac {e^{-\frac {4 \log ^2(5)}{\log (3)}} \text {Ei}\left (-\frac {x \log (3)-4 \log ^2(5)}{\log (3)}\right ) \log (3)}{4 \log ^2(5)}-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}-\left (4 \log ^2(5)\right ) \int \left (\frac {e^{-x} \log (3)}{16 x \log ^4(5)}+\frac {e^{-x}}{4 x^2 \log ^2(5)}-\frac {e^{-x} \log ^2(3)}{16 \log ^4(5) \left (x \log (3)-4 \log ^2(5)\right )}\right ) \, dx-\int \frac {e^{-x}}{x} \, dx\\ &=-\frac {e^{-x}}{x}-\text {Ei}(-x)+\frac {\text {Ei}(-x) \log (3)}{4 \log ^2(5)}-\frac {e^{-\frac {4 \log ^2(5)}{\log (3)}} \text {Ei}\left (-\frac {x \log (3)-4 \log ^2(5)}{\log (3)}\right ) \log (3)}{4 \log ^2(5)}-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}-\frac {\log (3) \int \frac {e^{-x}}{x} \, dx}{4 \log ^2(5)}+\frac {\log ^2(3) \int \frac {e^{-x}}{x \log (3)-4 \log ^2(5)} \, dx}{4 \log ^2(5)}-\int \frac {e^{-x}}{x^2} \, dx\\ &=-\text {Ei}(-x)-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}+\int \frac {e^{-x}}{x} \, dx\\ &=-\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 5.04, size = 25, normalized size = 0.78 \begin {gather*} -\frac {e^{-x} \log \left (-\log (3)+\frac {4 \log ^2(5)}{x}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 26, normalized size = 0.81 \begin {gather*} -\frac {e^{\left (-x\right )} \log \left (\frac {4 \, \log \relax (5)^{2} - x \log \relax (3)}{x}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 32, normalized size = 1.00 \begin {gather*} -\frac {e^{\left (-x\right )} \log \left (4 \, \log \relax (5)^{2} - x \log \relax (3)\right ) - e^{\left (-x\right )} \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 27, normalized size = 0.84
method | result | size |
norman | \(-\frac {\ln \left (\frac {4 \ln \relax (5)^{2}-x \ln \relax (3)}{x}\right ) {\mathrm e}^{-x}}{x}\) | \(27\) |
risch | \(-\frac {{\mathrm e}^{-x} \ln \left (\ln \relax (5)^{2}-\frac {x \ln \relax (3)}{4}\right )}{x}-\frac {\left (-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (-\ln \relax (5)^{2}+\frac {x \ln \relax (3)}{4}\right )}{x}\right )^{3}+4 \ln \relax (2)-2 \ln \relax (x )\right ) {\mathrm e}^{-x}}{2 x}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 32, normalized size = 1.00 \begin {gather*} -\frac {e^{\left (-x\right )} \log \left (4 \, \log \relax (5)^{2} - x \log \relax (3)\right ) - e^{\left (-x\right )} \log \relax (x)}{x} \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 = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-x}\,\left (\ln \left (-\frac {x\,\ln \relax (3)-4\,{\ln \relax (5)}^2}{x}\right )\,\left ({\ln \relax (5)}^2\,\left (4\,x+4\right )-\ln \relax (3)\,\left (x^2+x\right )\right )+4\,{\ln \relax (5)}^2\right )}{4\,x^2\,{\ln \relax (5)}^2-x^3\,\ln \relax (3)} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 20, normalized size = 0.62 \begin {gather*} - \frac {e^{- x} \log {\left (\frac {- x \log {\relax (3 )} + 4 \log {\relax (5 )}^{2}}{x} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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