Optimal. Leaf size=26 \[ \frac {e^{-x} \log (5)}{(x+\log (4)) \left (\log (4)+\frac {\log (\log (4))}{x}\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(26)=52\).
time = 1.09, antiderivative size = 63, normalized size of antiderivative = 2.42, number of steps
used = 10, number of rules used = 5, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6820, 12,
6874, 2208, 2209} \begin {gather*} \frac {e^{-x} \log (4) \log (5)}{\left (\log ^2(4)-\log (\log (4))\right ) (x+\log (4))}-\frac {e^{-x} \log (5) \log (\log (4))}{\left (\log ^2(4)-\log (\log (4))\right ) (x \log (4)+\log (\log (4)))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2208
Rule 2209
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \log (5) \left (-x^3 \log (4)+\log (4) \log (\log (4))-x \log (4) \log (\log (4))-x^2 \left (\log (4)+\log ^2(4)+\log (\log (4))\right )\right )}{(x+\log (4))^2 (x \log (4)+\log (\log (4)))^2} \, dx\\ &=\log (5) \int \frac {e^{-x} \left (-x^3 \log (4)+\log (4) \log (\log (4))-x \log (4) \log (\log (4))-x^2 \left (\log (4)+\log ^2(4)+\log (\log (4))\right )\right )}{(x+\log (4))^2 (x \log (4)+\log (\log (4)))^2} \, dx\\ &=\log (5) \int \left (-\frac {e^{-x} \log (4)}{(x+\log (4))^2 \left (\log ^2(4)-\log (\log (4))\right )}-\frac {e^{-x} \log (4)}{(x+\log (4)) \left (\log ^2(4)-\log (\log (4))\right )}+\frac {e^{-x} \log (4) \log (\log (4))}{\left (\log ^2(4)-\log (\log (4))\right ) (x \log (4)+\log (\log (4)))^2}+\frac {e^{-x} \log (\log (4))}{\left (\log ^2(4)-\log (\log (4))\right ) (x \log (4)+\log (\log (4)))}\right ) \, dx\\ &=-\frac {(\log (4) \log (5)) \int \frac {e^{-x}}{(x+\log (4))^2} \, dx}{\log ^2(4)-\log (\log (4))}-\frac {(\log (4) \log (5)) \int \frac {e^{-x}}{x+\log (4)} \, dx}{\log ^2(4)-\log (\log (4))}+\frac {(\log (5) \log (\log (4))) \int \frac {e^{-x}}{x \log (4)+\log (\log (4))} \, dx}{\log ^2(4)-\log (\log (4))}+\frac {(\log (4) \log (5) \log (\log (4))) \int \frac {e^{-x}}{(x \log (4)+\log (\log (4)))^2} \, dx}{\log ^2(4)-\log (\log (4))}\\ &=-\frac {4 \text {Ei}(-x-\log (4)) \log (4) \log (5)}{\log ^2(4)-\log (\log (4))}+\frac {e^{-x} \log (4) \log (5)}{(x+\log (4)) \left (\log ^2(4)-\log (\log (4))\right )}+\frac {\text {Ei}\left (-\frac {x \log (4)+\log (\log (4))}{\log (4)}\right ) \log ^{-1+\frac {1}{\log (4)}}(4) \log (5) \log (\log (4))}{\log ^2(4)-\log (\log (4))}-\frac {e^{-x} \log (5) \log (\log (4))}{\left (\log ^2(4)-\log (\log (4))\right ) (x \log (4)+\log (\log (4)))}+\frac {(\log (4) \log (5)) \int \frac {e^{-x}}{x+\log (4)} \, dx}{\log ^2(4)-\log (\log (4))}-\frac {(\log (5) \log (\log (4))) \int \frac {e^{-x}}{x \log (4)+\log (\log (4))} \, dx}{\log ^2(4)-\log (\log (4))}\\ &=\frac {e^{-x} \log (4) \log (5)}{(x+\log (4)) \left (\log ^2(4)-\log (\log (4))\right )}-\frac {e^{-x} \log (5) \log (\log (4))}{\left (\log ^2(4)-\log (\log (4))\right ) (x \log (4)+\log (\log (4)))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 25, normalized size = 0.96 \begin {gather*} \frac {e^{-x} x \log (5)}{(x+\log (4)) (x \log (4)+\log (\log (4)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(556\) vs.
\(2(31)=62\).
time = 0.79, size = 557, normalized size = 21.42
method | result | size |
gosper | \(\frac {x \ln \left (5\right ) {\mathrm e}^{-x}}{4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2 \ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2 \ln \left (2\right )\right )}\) | \(42\) |
norman | \(\frac {x \ln \left (5\right ) {\mathrm e}^{-x}}{4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2 \ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2 \ln \left (2\right )\right )}\) | \(42\) |
risch | \(\frac {x \ln \left (5\right ) {\mathrm e}^{-x}}{4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )}\) | \(48\) |
default | \(\frac {2 \ln \left (2\right ) \ln \left (5\right ) {\mathrm e}^{-x} x \ln \left (\ln \left (2\right )\right )}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}-\frac {8 \ln \left (2\right )^{2} \ln \left (5\right ) {\mathrm e}^{-x} x \ln \left (\ln \left (2\right )\right )}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}+\frac {\ln \left (2\right )^{2} \ln \left (5\right ) {\mathrm e}^{-x} x}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}+\frac {\ln \left (5\right ) {\mathrm e}^{-x} \ln \left (\ln \left (2\right )\right )^{2} x}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}+\frac {16 \ln \left (2\right )^{4} \ln \left (5\right ) {\mathrm e}^{-x} x}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}-\frac {8 \ln \left (2\right )^{3} \ln \left (5\right ) {\mathrm e}^{-x} x}{\left (16 \ln \left (2\right )^{4}-8 \ln \left (2\right )^{3}-8 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )+\ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) \left (4 x \ln \left (2\right )^{2}+2 x^{2} \ln \left (2\right )+2 \ln \left (2\right )^{2}+2 \ln \left (\ln \left (2\right )\right ) \ln \left (2\right )+x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )\right )}\) | \(557\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.69, size = 45, normalized size = 1.73 \begin {gather*} \frac {x e^{\left (-x\right )} \log \left (5\right )}{2 \, x^{2} \log \left (2\right ) + {\left (4 \, \log \left (2\right )^{2} + \log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} x + 2 \, \log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 39, normalized size = 1.50 \begin {gather*} \frac {x \log \left (5\right )}{{\left (x + 2 \, \log \left (2\right )\right )} e^{x} \log \left (2 \, \log \left (2\right )\right ) + 2 \, {\left (x^{2} \log \left (2\right ) + 2 \, x \log \left (2\right )^{2}\right )} e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (26) = 52\).
time = 0.17, size = 53, normalized size = 2.04 \begin {gather*} \frac {x e^{- x} \log {\left (5 \right )}}{2 x^{2} \log {\left (2 \right )} + x \log {\left (\log {\left (2 \right )} \right )} + x \log {\left (2 \right )} + 4 x \log {\left (2 \right )}^{2} + 2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + 2 \log {\left (2 \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 47, normalized size = 1.81 \begin {gather*} \frac {x e^{\left (-x\right )} \log \left (5\right )}{2 \, x^{2} \log \left (2\right ) + 4 \, x \log \left (2\right )^{2} + x \log \left (2\right ) + 2 \, \log \left (2\right )^{2} + x \log \left (\log \left (2\right )\right ) + 2 \, \log \left (2\right ) \log \left (\log \left (2\right )\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 {\ln \left (5\right )\,\left (4\,x^2\,{\ln \left (2\right )}^2+2\,\ln \left (2\right )\,\left (x^3+x^2\right )\right )+\ln \left (2\,\ln \left (2\right )\right )\,\ln \left (5\right )\,\left (2\,\ln \left (2\right )\,\left (x-1\right )+x^2\right )}{{\mathrm {e}}^x\,\left (4\,{\ln \left (2\right )}^2\,x^4+16\,{\ln \left (2\right )}^3\,x^3+16\,{\ln \left (2\right )}^4\,x^2\right )+\ln \left (2\,\ln \left (2\right )\right )\,{\mathrm {e}}^x\,\left (4\,\ln \left (2\right )\,x^3+16\,{\ln \left (2\right )}^2\,x^2+16\,{\ln \left (2\right )}^3\,x\right )+{\ln \left (2\,\ln \left (2\right )\right )}^2\,{\mathrm {e}}^x\,\left (x^2+4\,\ln \left (2\right )\,x+4\,{\ln \left (2\right )}^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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