Optimal. Leaf size=20 \[ \frac {\left (e^x+\frac {2+\log (\log (2))}{x}\right ) \log (\log (x))}{x^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.81, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 6, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6741, 6742, 2309, 2178, 2522, 2288} \begin {gather*} \frac {(2+\log (\log (2))) \log (\log (x))}{x^3}+\frac {e^x \log (\log (x))}{x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2288
Rule 2309
Rule 2522
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x x+2 \left (1+\frac {1}{2} \log (\log (2))\right )+\left (\left (-6+e^x \left (-2 x+x^2\right )\right ) \log (x)-3 \log (x) \log (\log (2))\right ) \log (\log (x))}{x^4 \log (x)} \, dx\\ &=\int \left (-\frac {(2+\log (\log (2))) (-1+3 \log (x) \log (\log (x)))}{x^4 \log (x)}+\frac {e^x (1-2 \log (x) \log (\log (x))+x \log (x) \log (\log (x)))}{x^3 \log (x)}\right ) \, dx\\ &=(-2-\log (\log (2))) \int \frac {-1+3 \log (x) \log (\log (x))}{x^4 \log (x)} \, dx+\int \frac {e^x (1-2 \log (x) \log (\log (x))+x \log (x) \log (\log (x)))}{x^3 \log (x)} \, dx\\ &=\frac {e^x \log (\log (x))}{x^2}+(-2-\log (\log (2))) \int \left (-\frac {1}{x^4 \log (x)}+\frac {3 \log (\log (x))}{x^4}\right ) \, dx\\ &=\frac {e^x \log (\log (x))}{x^2}+(2+\log (\log (2))) \int \frac {1}{x^4 \log (x)} \, dx-(3 (2+\log (\log (2)))) \int \frac {\log (\log (x))}{x^4} \, dx\\ &=\frac {e^x \log (\log (x))}{x^2}+\frac {(2+\log (\log (2))) \log (\log (x))}{x^3}-(2+\log (\log (2))) \int \frac {1}{x^4 \log (x)} \, dx+(2+\log (\log (2))) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{x} \, dx,x,\log (x)\right )\\ &=\text {Ei}(-3 \log (x)) (2+\log (\log (2)))+\frac {e^x \log (\log (x))}{x^2}+\frac {(2+\log (\log (2))) \log (\log (x))}{x^3}-(2+\log (\log (2))) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {e^x \log (\log (x))}{x^2}+\frac {(2+\log (\log (2))) \log (\log (x))}{x^3}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 17, normalized size = 0.85 \begin {gather*} \frac {\left (2+e^x x+\log (\log (2))\right ) \log (\log (x))}{x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.40, size = 16, normalized size = 0.80 \begin {gather*} \frac {{\left (x e^{x} + \log \left (\log \relax (2)\right ) + 2\right )} \log \left (\log \relax (x)\right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 24, normalized size = 1.20 \begin {gather*} \frac {x e^{x} \log \left (\log \relax (x)\right ) + \log \left (\log \relax (2)\right ) \log \left (\log \relax (x)\right ) + 2 \, \log \left (\log \relax (x)\right )}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 17, normalized size = 0.85
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{x} x +\ln \left (\ln \relax (2)\right )+2\right ) \ln \left (\ln \relax (x )\right )}{x^{3}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -{\left (\log \left (\log \relax (2)\right ) + 2\right )} \int \frac {1}{x^{4} \log \relax (x)}\,{d x} + {\rm Ei}\left (-3 \, \log \relax (x)\right ) \log \left (\log \relax (2)\right ) + \frac {{\left (x e^{x} + \log \left (\log \relax (2)\right ) + 2\right )} \log \left (\log \relax (x)\right )}{x^{3}} + 2 \, {\rm Ei}\left (-3 \, \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\ln \left (\ln \relax (2)\right )-\ln \left (\ln \relax (x)\right )\,\left (3\,\ln \left (\ln \relax (2)\right )\,\ln \relax (x)+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (2\,x-x^2\right )+6\right )\right )+x\,{\mathrm {e}}^x+2}{x^4\,\ln \relax (x)} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.46, size = 26, normalized size = 1.30 \begin {gather*} \frac {e^{x} \log {\left (\log {\relax (x )} \right )}}{x^{2}} + \frac {\left (\log {\left (\log {\relax (2 )} \right )} + 2\right ) \log {\left (\log {\relax (x )} \right )}}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________