Optimal. Leaf size=33 \[ \left (5+e^{2 x}\right ) \left (-x+e^4 x-\log (2+x)\right ) \log (\log (i \pi +\log (2))) \]
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Rubi [C] time = 0.77, antiderivative size = 124, normalized size of antiderivative = 3.76, number of steps used = 16, number of rules used = 9, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 6742, 43, 6688, 2199, 2194, 2178, 2176, 2554} \begin {gather*} -\frac {\log (\log (\log (2)+i \pi )) \text {Ei}(2 (x+2))}{e^4}+\frac {\log (\log (\log (2)+i \pi )) \text {Ei}(2 x+4)}{e^4}+\left (1-e^4\right ) \left (-e^{2 x}\right ) x \log (\log (\log (2)+i \pi ))-5 \left (1-e^4\right ) x \log (\log (\log (2)+i \pi ))-e^{2 x} \log (\log (\log (2)+i \pi )) \log (x+2)-5 \log (\log (\log (2)+i \pi )) \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2176
Rule 2178
Rule 2194
Rule 2199
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log (\log (i \pi +\log (2))) \int \frac {-15-5 x+e^4 (10+5 x)+e^{2 x} \left (-3-5 x-2 x^2+e^4 \left (2+5 x+2 x^2\right )\right )+e^{2 x} (-4-2 x) \log (2+x)}{2+x} \, dx\\ &=\log (\log (i \pi +\log (2))) \int \left (\frac {5 \left (-3+2 e^4-\left (1-e^4\right ) x\right )}{2+x}+\frac {e^{2 x} \left (-3 \left (1-\frac {2 e^4}{3}\right )-5 \left (1-e^4\right ) x-2 \left (1-e^4\right ) x^2-4 \log (2+x)-2 x \log (2+x)\right )}{2+x}\right ) \, dx\\ &=\log (\log (i \pi +\log (2))) \int \frac {e^{2 x} \left (-3 \left (1-\frac {2 e^4}{3}\right )-5 \left (1-e^4\right ) x-2 \left (1-e^4\right ) x^2-4 \log (2+x)-2 x \log (2+x)\right )}{2+x} \, dx+(5 \log (\log (i \pi +\log (2)))) \int \frac {-3+2 e^4-\left (1-e^4\right ) x}{2+x} \, dx\\ &=\log (\log (i \pi +\log (2))) \int \frac {e^{2 x} \left (-3-5 x-2 x^2+e^4 \left (2+5 x+2 x^2\right )-2 (2+x) \log (2+x)\right )}{2+x} \, dx+(5 \log (\log (i \pi +\log (2)))) \int \left (-1+e^4+\frac {1}{-2-x}\right ) \, dx\\ &=-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-5 \log (2+x) \log (\log (i \pi +\log (2)))+\log (\log (i \pi +\log (2))) \int \left (\frac {e^{2 x} \left (-3+2 e^4-5 \left (1-e^4\right ) x-2 \left (1-e^4\right ) x^2\right )}{2+x}-2 e^{2 x} \log (2+x)\right ) \, dx\\ &=-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-5 \log (2+x) \log (\log (i \pi +\log (2)))+\log (\log (i \pi +\log (2))) \int \frac {e^{2 x} \left (-3+2 e^4-5 \left (1-e^4\right ) x-2 \left (1-e^4\right ) x^2\right )}{2+x} \, dx-(2 \log (\log (i \pi +\log (2)))) \int e^{2 x} \log (2+x) \, dx\\ &=-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-5 \log (2+x) \log (\log (i \pi +\log (2)))-e^{2 x} \log (2+x) \log (\log (i \pi +\log (2)))+\log (\log (i \pi +\log (2))) \int \left (e^{2 x} \left (-1+e^4\right )+\frac {e^{2 x}}{-2-x}+2 e^{2 x} \left (-1+e^4\right ) x\right ) \, dx+(2 \log (\log (i \pi +\log (2)))) \int \frac {e^{2 x}}{4+2 x} \, dx\\ &=-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))+\frac {\text {Ei}(4+2 x) \log (\log (i \pi +\log (2)))}{e^4}-5 \log (2+x) \log (\log (i \pi +\log (2)))-e^{2 x} \log (2+x) \log (\log (i \pi +\log (2)))+\log (\log (i \pi +\log (2))) \int \frac {e^{2 x}}{-2-x} \, dx-\left (\left (1-e^4\right ) \log (\log (i \pi +\log (2)))\right ) \int e^{2 x} \, dx-\left (2 \left (1-e^4\right ) \log (\log (i \pi +\log (2)))\right ) \int e^{2 x} x \, dx\\ &=-\frac {1}{2} e^{2 x} \left (1-e^4\right ) \log (\log (i \pi +\log (2)))-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-e^{2 x} \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-\frac {\text {Ei}(2 (2+x)) \log (\log (i \pi +\log (2)))}{e^4}+\frac {\text {Ei}(4+2 x) \log (\log (i \pi +\log (2)))}{e^4}-5 \log (2+x) \log (\log (i \pi +\log (2)))-e^{2 x} \log (2+x) \log (\log (i \pi +\log (2)))+\left (\left (1-e^4\right ) \log (\log (i \pi +\log (2)))\right ) \int e^{2 x} \, dx\\ &=-5 \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-e^{2 x} \left (1-e^4\right ) x \log (\log (i \pi +\log (2)))-\frac {\text {Ei}(2 (2+x)) \log (\log (i \pi +\log (2)))}{e^4}+\frac {\text {Ei}(4+2 x) \log (\log (i \pi +\log (2)))}{e^4}-5 \log (2+x) \log (\log (i \pi +\log (2)))-e^{2 x} \log (2+x) \log (\log (i \pi +\log (2)))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.40, size = 42, normalized size = 1.27 \begin {gather*} \left (\left (-1+e^4\right ) \left (10+\left (5+e^{2 x}\right ) x\right )-\left (5+e^{2 x}\right ) \log (2+x)\right ) \log (\log (i \pi +\log (2))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 43, normalized size = 1.30 \begin {gather*} {\left (5 \, x e^{4} + {\left (x e^{4} - x\right )} e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 5\right )} \log \left (x + 2\right ) - 5 \, x\right )} \log \left (\log \left (i \, \pi + \log \relax (2)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 49, normalized size = 1.48 \begin {gather*} {\left (5 \, x e^{4} - x e^{\left (2 \, x\right )} + x e^{\left (2 \, x + 4\right )} - e^{\left (2 \, x\right )} \log \left (x + 2\right ) - 5 \, x - 5 \, \log \left (x + 2\right )\right )} \log \left (\log \left (i \, \pi + \log \relax (2)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 46, normalized size = 1.39
method | result | size |
default | \(\ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right ) \left (\left ({\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} \ln \left (2+x \right )+5 x \,{\mathrm e}^{4}-5 x -5 \ln \left (2+x \right )\right )\) | \(46\) |
norman | \(\left (5 \ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right ) {\mathrm e}^{4}-5 \ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right )\right ) x -5 \ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right ) \ln \left (2+x \right )+\left (\ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right ) {\mathrm e}^{4}-\ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right )\right ) x \,{\mathrm e}^{2 x}-\ln \left (\ln \left (\ln \relax (2)+i \pi \right )\right ) {\mathrm e}^{2 x} \ln \left (2+x \right )\) | \(93\) |
risch | \(-\ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) {\mathrm e}^{2 x} \ln \left (2+x \right )+\ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) x \,{\mathrm e}^{2 x +4}+5 \ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) x \,{\mathrm e}^{4}-\ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) x \,{\mathrm e}^{2 x}-5 \ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) \ln \left (2+x \right )-5 \ln \left (\ln \left (-i \left (i \ln \relax (2)-\pi \right )\right )\right ) x\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} {\left (x {\left (e^{4} - 1\right )} e^{\left (2 \, x\right )} + 5 \, {\left (x - 2 \, \log \left (x + 2\right )\right )} e^{4} + 3 \, e^{\left (-4\right )} E_{1}\left (-2 \, x - 4\right ) + 10 \, e^{4} \log \left (x + 2\right ) - e^{\left (2 \, x\right )} \log \left (x + 2\right ) - 5 \, x + 3 \, \int \frac {e^{\left (2 \, x\right )}}{x + 2}\,{d x} - 5 \, \log \left (x + 2\right )\right )} \log \left (\log \left (i \, \pi + \log \relax (2)\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 50, normalized size = 1.52 \begin {gather*} -\ln \left (\ln \left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )\right )\,\left (5\,x+5\,\ln \left (x+2\right )+x\,{\mathrm {e}}^{2\,x}-5\,x\,{\mathrm {e}}^4-x\,{\mathrm {e}}^{2\,x+4}+\ln \left (x+2\right )\,{\mathrm {e}}^{2\,x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.70, size = 92, normalized size = 2.79 \begin {gather*} x \left (- 5 \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )} + 5 e^{4} \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )}\right ) + \left (- x \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )} + x e^{4} \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )} - \log {\left (x + 2 \right )} \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )}\right ) e^{2 x} - 5 \log {\left (x + 2 \right )} \log {\left (\log {\left (\log {\relax (2 )} + i \pi \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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