Optimal. Leaf size=22 \[ 1+x+\frac {4 e^{e^{-1+e^{e^x}}}}{(x+\log (x))^4} \]
________________________________________________________________________________________
Rubi [F] time = 3.97, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)+e^{e^{-1+e^{e^x}}} \left (-16-16 x+e^{-1+e^{e^x}+e^x} \left (4 e^x x^2+4 e^x x \log (x)\right )\right )}{x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)+e^{e^{-1+e^{e^x}}} \left (-16-16 x+e^{-1+e^{e^x}+e^x} \left (4 e^x x^2+4 e^x x \log (x)\right )\right )}{x (x+\log (x))^5} \, dx\\ &=\int \left (-\frac {16 e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5}-\frac {16 e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5}+\frac {x^5}{(x+\log (x))^5}+\frac {5 x^4 \log (x)}{(x+\log (x))^5}+\frac {10 x^3 \log ^2(x)}{(x+\log (x))^5}+\frac {10 x^2 \log ^3(x)}{(x+\log (x))^5}+\frac {5 x \log ^4(x)}{(x+\log (x))^5}+\frac {\log ^5(x)}{(x+\log (x))^5}+\frac {4 e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4}\right ) \, dx\\ &=4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+5 \int \frac {x^4 \log (x)}{(x+\log (x))^5} \, dx+5 \int \frac {x \log ^4(x)}{(x+\log (x))^5} \, dx+10 \int \frac {x^3 \log ^2(x)}{(x+\log (x))^5} \, dx+10 \int \frac {x^2 \log ^3(x)}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx+\int \frac {x^5}{(x+\log (x))^5} \, dx+\int \frac {\log ^5(x)}{(x+\log (x))^5} \, dx\\ &=4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+5 \int \left (-\frac {x^5}{(x+\log (x))^5}+\frac {x^4}{(x+\log (x))^4}\right ) \, dx+5 \int \left (\frac {x^5}{(x+\log (x))^5}-\frac {4 x^4}{(x+\log (x))^4}+\frac {6 x^3}{(x+\log (x))^3}-\frac {4 x^2}{(x+\log (x))^2}+\frac {x}{x+\log (x)}\right ) \, dx+10 \int \left (\frac {x^5}{(x+\log (x))^5}-\frac {2 x^4}{(x+\log (x))^4}+\frac {x^3}{(x+\log (x))^3}\right ) \, dx+10 \int \left (-\frac {x^5}{(x+\log (x))^5}+\frac {3 x^4}{(x+\log (x))^4}-\frac {3 x^3}{(x+\log (x))^3}+\frac {x^2}{(x+\log (x))^2}\right ) \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx+\int \frac {x^5}{(x+\log (x))^5} \, dx+\int \left (1-\frac {x^5}{(x+\log (x))^5}+\frac {5 x^4}{(x+\log (x))^4}-\frac {10 x^3}{(x+\log (x))^3}+\frac {10 x^2}{(x+\log (x))^2}-\frac {5 x}{x+\log (x)}\right ) \, dx\\ &=x+4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+2 \left (5 \int \frac {x^4}{(x+\log (x))^4} \, dx\right )+2 \left (10 \int \frac {x^2}{(x+\log (x))^2} \, dx\right )-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx-2 \left (20 \int \frac {x^4}{(x+\log (x))^4} \, dx\right )-20 \int \frac {x^2}{(x+\log (x))^2} \, dx+30 \int \frac {x^4}{(x+\log (x))^4} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 29, normalized size = 1.32 \begin {gather*} \frac {e x+\frac {4 e^{1+e^{-1+e^{e^x}}}}{(x+\log (x))^4}}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.83, size = 78, normalized size = 3.55 \begin {gather*} \frac {x^{5} + 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + 4 \, e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{4} + 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 5 \, x^{5} \log \relax (x) + 10 \, x^{4} \log \relax (x)^{2} + 10 \, x^{3} \log \relax (x)^{3} + 5 \, x^{2} \log \relax (x)^{4} + x \log \relax (x)^{5} + 4 \, {\left ({\left (x^{2} e^{x} + x e^{x} \log \relax (x)\right )} e^{\left (e^{x} + e^{\left (e^{x}\right )} - 1\right )} - 4 \, x - 4\right )} e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{6} + 5 \, x^{5} \log \relax (x) + 10 \, x^{4} \log \relax (x)^{2} + 10 \, x^{3} \log \relax (x)^{3} + 5 \, x^{2} \log \relax (x)^{4} + x \log \relax (x)^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 45, normalized size = 2.05
method | result | size |
risch | \(x +\frac {4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}-1}}}{x^{4}+4 x^{3} \ln \relax (x )+6 x^{2} \ln \relax (x )^{2}+4 x \ln \relax (x )^{3}+\ln \relax (x )^{4}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 78, normalized size = 3.55 \begin {gather*} \frac {x^{5} + 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + 4 \, e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{4} + 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.41, size = 18, normalized size = 0.82 \begin {gather*} x+\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^{-1}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}}{{\left (x+\ln \relax (x)\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.62, size = 48, normalized size = 2.18 \begin {gather*} x + \frac {4 e^{e^{e^{e^{x}} - 1}}}{x^{4} + 4 x^{3} \log {\relax (x )} + 6 x^{2} \log {\relax (x )}^{2} + 4 x \log {\relax (x )}^{3} + \log {\relax (x )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________