Optimal. Leaf size=27 \[ e^x (i \pi +\log (2)) \left (1+\log \left (4 e^{2/x}\right ) \log (x)\right ) \]
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Rubi [A] time = 0.59, antiderivative size = 37, normalized size of antiderivative = 1.37, number of steps used = 29, number of rules used = 10, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {14, 2194, 2178, 2554, 12, 6485, 2177, 6483, 6475, 2557} \begin {gather*} e^x (\log (2)+i \pi ) \log \left (4 e^{2/x}\right ) \log (x)+e^x (\log (2)+i \pi ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2177
Rule 2178
Rule 2194
Rule 2554
Rule 2557
Rule 6475
Rule 6483
Rule 6485
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x (i \pi +\log (2))+\frac {e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right )}{x}-\frac {2 i e^x (\pi -i \log (2)) \log (x)}{x^2}+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)\right ) \, dx\\ &=-\left ((2 i (\pi -i \log (2))) \int \frac {e^x \log (x)}{x^2} \, dx\right )+(i \pi +\log (2)) \int e^x \, dx+(i \pi +\log (2)) \int \frac {e^x \log \left (4 e^{2/x}\right )}{x} \, dx+(i \pi +\log (2)) \int e^x \log \left (4 e^{2/x}\right ) \log (x) \, dx\\ &=e^x (i \pi +\log (2))+\text {Ei}(x) (i \pi +\log (2)) \log \left (4 e^{2/x}\right )+\frac {2 e^x (i \pi +\log (2)) \log (x)}{x}-2 \text {Ei}(x) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(-i \pi -\log (2)) \int -\frac {2 \text {Ei}(x)}{x^2} \, dx+(-i \pi -\log (2)) \int \frac {e^x \log \left (4 e^{2/x}\right )}{x} \, dx+(-i \pi -\log (2)) \int -\frac {2 e^x \log (x)}{x^2} \, dx+(2 i (\pi -i \log (2))) \int \frac {-e^x+x \text {Ei}(x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))+\frac {2 e^x (i \pi +\log (2)) \log (x)}{x}-2 \text {Ei}(x) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 i (\pi -i \log (2))) \int \left (-\frac {e^x}{x^2}+\frac {\text {Ei}(x)}{x}\right ) \, dx+(i \pi +\log (2)) \int -\frac {2 \text {Ei}(x)}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x \log (x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))-\frac {2 \text {Ei}(x) (i \pi +\log (2))}{x}+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)-(2 i (\pi -i \log (2))) \int \frac {e^x}{x^2} \, dx+(2 i (\pi -i \log (2))) \int \frac {\text {Ei}(x)}{x} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {-e^x+x \text {Ei}(x)}{x^2} \, dx\\ &=e^x (i \pi +\log (2))+2 (E_1(-x)+\text {Ei}(x)) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)-(2 i (\pi -i \log (2))) \int \frac {e^x}{x} \, dx-(2 i (\pi -i \log (2))) \int \frac {E_1(-x)}{x} \, dx-(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx+(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx-(2 (i \pi +\log (2))) \int \left (-\frac {e^x}{x^2}+\frac {\text {Ei}(x)}{x}\right ) \, dx\\ &=e^x (i \pi +\log (2))+\frac {2 e^x (i \pi +\log (2))}{x}+2 x \, _3F_3(1,1,1;2,2,2;x) (i \pi +\log (2))+(i \pi +\log (2)) \log ^2(-x)+2 \gamma (i \pi +\log (2)) \log (x)+2 (E_1(-x)+\text {Ei}(x)) (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 (i \pi +\log (2))) \int \frac {e^x}{x^2} \, dx-(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx-(2 (i \pi +\log (2))) \int \frac {\text {Ei}(x)}{x} \, dx\\ &=e^x (i \pi +\log (2))-2 \text {Ei}(x) (i \pi +\log (2))+2 x \, _3F_3(1,1,1;2,2,2;x) (i \pi +\log (2))+(i \pi +\log (2)) \log ^2(-x)+2 \gamma (i \pi +\log (2)) \log (x)+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)+(2 (i \pi +\log (2))) \int \frac {e^x}{x} \, dx+(2 (i \pi +\log (2))) \int \frac {E_1(-x)}{x} \, dx\\ &=e^x (i \pi +\log (2))+e^x (i \pi +\log (2)) \log \left (4 e^{2/x}\right ) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 27, normalized size = 1.00 \begin {gather*} e^x (i \pi +\log (2)) \left (1+\log \left (4 e^{2/x}\right ) \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 45, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (-i \, \pi - x \log \relax (2)^{2} + {\left (-i \, \pi x - 1\right )} \log \relax (2)\right )} e^{x} \log \relax (x) - {\left (i \, \pi x + x \log \relax (2)\right )} e^{x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 53, normalized size = 1.96 \begin {gather*} \frac {2 i \, \pi x e^{x} \log \relax (2) \log \relax (x) + 2 \, x e^{x} \log \relax (2)^{2} \log \relax (x) + i \, \pi x e^{x} + x e^{x} \log \relax (2) + 2 i \, \pi e^{x} \log \relax (x) + 2 \, e^{x} \log \relax (2) \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 52, normalized size = 1.93
method | result | size |
risch | \(\left (\ln \relax (2)+i \pi \right ) {\mathrm e}^{x} \ln \relax (x ) \ln \left ({\mathrm e}^{\frac {2}{x}}\right )+2 i \pi \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )+2 \ln \relax (2)^{2} {\mathrm e}^{x} \ln \relax (x )+i \pi \,{\mathrm e}^{x}+{\mathrm e}^{x} \ln \relax (2)\) | \(52\) |
default | \(\frac {x \left (\ln \relax (2)+i \pi \right ) {\mathrm e}^{x}+\left (2 \ln \relax (2)+2 i \pi \right ) {\mathrm e}^{x} \ln \relax (x )+\left (\ln \relax (2) \left (\ln \left (4 \,{\mathrm e}^{\frac {2}{x}}\right )-\frac {2}{x}\right )+i \pi \left (\ln \left (4 \,{\mathrm e}^{\frac {2}{x}}\right )-\frac {2}{x}\right )\right ) x \,{\mathrm e}^{x} \ln \relax (x )}{x}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 42, normalized size = 1.56 \begin {gather*} i \, \pi e^{x} + e^{x} \log \relax (2) - \frac {2 \, {\left (-i \, \pi + {\left (-i \, \pi \log \relax (2) - \log \relax (2)^{2}\right )} x - \log \relax (2)\right )} e^{x} \log \relax (x)}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.50, size = 26, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^x\,\left (\ln \relax (2)+\Pi \,1{}\mathrm {i}\right )\,\left (x+2\,\ln \relax (x)+2\,x\,\ln \relax (2)\,\ln \relax (x)\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.64, size = 60, normalized size = 2.22 \begin {gather*} - \frac {\left (- 2 x \log {\relax (2 )}^{2} \log {\relax (x )} - 2 i \pi x \log {\relax (2 )} \log {\relax (x )} - x \log {\relax (2 )} - i \pi x - 2 \log {\relax (2 )} \log {\relax (x )} - 2 i \pi \log {\relax (x )}\right ) e^{x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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