Integrand size = 118, antiderivative size = 18 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=e^{\left (3+\frac {x}{2}+x^{-2+x}\right )^2}+x \] Output:
exp((1/2*x+exp(x*ln(x))/x^2+3)^2)+x
Time = 2.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=e^{9+3 x+\frac {x^2}{4}+x^{-4+2 x}+x^{-2+x} (6+x)}+x \] Input:
Integrate[(2*x^5 + E^((36*x^4 + 12*x^5 + x^6 + 4*x^(2*x) + x^x*(24*x^2 + 4 *x^3))/(4*x^4))*(6*x^5 + x^6 + x^(2*x)*(-8 + 4*x + 4*x*Log[x]) + x^x*(-24* x^2 + 10*x^3 + 2*x^4 + (12*x^3 + 2*x^4)*Log[x])))/(2*x^5),x]
Output:
E^(9 + 3*x + x^2/4 + x^(-4 + 2*x) + x^(-2 + x)*(6 + x)) + x
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\exp \left (\frac {4 x^{2 x}+x^6+12 x^5+36 x^4+\left (4 x^3+24 x^2\right ) x^x}{4 x^4}\right ) \left (x^{2 x} (4 x+4 x \log (x)-8)+x^6+6 x^5+x^x \left (2 x^4+10 x^3-24 x^2+\left (2 x^4+12 x^3\right ) \log (x)\right )\right )+2 x^5}{2 x^5} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {2 x^5+\exp \left (\frac {4 \left (x^3+6 x^2\right ) x^x+4 x^{2 x}+x^6+12 x^5+36 x^4}{4 x^4}\right ) \left (-2 \left (-x^4-5 x^3+12 x^2-\left (x^4+6 x^3\right ) \log (x)\right ) x^x-4 (-\log (x) x-x+2) x^{2 x}+x^6+6 x^5\right )}{x^5}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{2} \int \left (2 e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} \left (\log (x) x^2+x^2+6 \log (x) x+5 x-12\right ) x^{x-3}+4 e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} (\log (x) x+x-2) x^{2 x-5}+e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x+6 e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (6 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}}dx+\int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} xdx-24 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-3}dx+10 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-2}dx+2 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-1}dx-8 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{2 x-5}dx+4 \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{2 x-4}dx-12 \int \frac {\int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-2}dx}{x}dx-2 \int \frac {\int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-1}dx}{x}dx-4 \int \frac {\int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{2 x-4}dx}{x}dx+12 \log (x) \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-2}dx+2 \log (x) \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{x-1}dx+4 \log (x) \int e^{\frac {\left (2 x^x+x^3+6 x^2\right )^2}{4 x^4}} x^{2 x-4}dx+2 x\right )\) |
Input:
Int[(2*x^5 + E^((36*x^4 + 12*x^5 + x^6 + 4*x^(2*x) + x^x*(24*x^2 + 4*x^3)) /(4*x^4))*(6*x^5 + x^6 + x^(2*x)*(-8 + 4*x + 4*x*Log[x]) + x^x*(-24*x^2 + 10*x^3 + 2*x^4 + (12*x^3 + 2*x^4)*Log[x])))/(2*x^5),x]
Output:
$Aborted
Time = 2.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39
method | result | size |
risch | \(x +{\mathrm e}^{\frac {\left (x^{3}+6 x^{2}+2 x^{x}\right )^{2}}{4 x^{4}}}\) | \(25\) |
parallelrisch | \(x +{\mathrm e}^{\frac {4 \,{\mathrm e}^{2 x \ln \left (x \right )}+\left (4 x^{3}+24 x^{2}\right ) {\mathrm e}^{x \ln \left (x \right )}+x^{6}+12 x^{5}+36 x^{4}}{4 x^{4}}}\) | \(49\) |
Input:
int(1/2*(((4*x*ln(x)+4*x-8)*exp(x*ln(x))^2+((2*x^4+12*x^3)*ln(x)+2*x^4+10* x^3-24*x^2)*exp(x*ln(x))+x^6+6*x^5)*exp(1/4*(4*exp(x*ln(x))^2+(4*x^3+24*x^ 2)*exp(x*ln(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x,method=_RETURNVERBOSE )
Output:
x+exp(1/4*(x^3+6*x^2+2*x^x)^2/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.39 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=x + e^{\left (\frac {x^{6} + 12 \, x^{5} + 36 \, x^{4} + 4 \, {\left (x^{3} + 6 \, x^{2}\right )} x^{x} + 4 \, x^{2 \, x}}{4 \, x^{4}}\right )} \] Input:
integrate(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+ 2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+( 4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algorith m="fricas")
Output:
x + e^(1/4*(x^6 + 12*x^5 + 36*x^4 + 4*(x^3 + 6*x^2)*x^x + 4*x^(2*x))/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.47 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=x + e^{\frac {\frac {x^{6}}{4} + 3 x^{5} + 9 x^{4} + \frac {\left (4 x^{3} + 24 x^{2}\right ) e^{x \log {\left (x \right )}}}{4} + e^{2 x \log {\left (x \right )}}}{x^{4}}} \] Input:
integrate(1/2*(((4*x*ln(x)+4*x-8)*exp(x*ln(x))**2+((2*x**4+12*x**3)*ln(x)+ 2*x**4+10*x**3-24*x**2)*exp(x*ln(x))+x**6+6*x**5)*exp(1/4*(4*exp(x*ln(x))* *2+(4*x**3+24*x**2)*exp(x*ln(x))+x**6+12*x**5+36*x**4)/x**4)+2*x**5)/x**5, x)
Output:
x + exp((x**6/4 + 3*x**5 + 9*x**4 + (4*x**3 + 24*x**2)*exp(x*log(x))/4 + e xp(2*x*log(x)))/x**4)
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=x + e^{\left (\frac {1}{4} \, x^{2} + 3 \, x + \frac {x^{x}}{x} + \frac {6 \, x^{x}}{x^{2}} + \frac {x^{2 \, x}}{x^{4}} + 9\right )} \] Input:
integrate(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+ 2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+( 4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algorith m="maxima")
Output:
x + e^(1/4*x^2 + 3*x + x^x/x + 6*x^x/x^2 + x^(2*x)/x^4 + 9)
Timed out. \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=\text {Timed out} \] Input:
integrate(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+ 2*x^4+10*x^3-24*x^2)*exp(x*log(x))+x^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+( 4*x^3+24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x, algorith m="giac")
Output:
Timed out
Time = 3.74 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=x+{\left ({\mathrm {e}}^{x^2}\right )}^{1/4}\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{\frac {x^{2\,x}}{x^4}}\,{\mathrm {e}}^9\,{\mathrm {e}}^{\frac {x^x}{x}}\,{\mathrm {e}}^{\frac {6\,x^x}{x^2}} \] Input:
int((x^5 + (exp((exp(2*x*log(x)) + (exp(x*log(x))*(24*x^2 + 4*x^3))/4 + 9* x^4 + 3*x^5 + x^6/4)/x^4)*(exp(2*x*log(x))*(4*x + 4*x*log(x) - 8) + exp(x* log(x))*(log(x)*(12*x^3 + 2*x^4) - 24*x^2 + 10*x^3 + 2*x^4) + 6*x^5 + x^6) )/2)/x^5,x)
Output:
x + exp(x^2)^(1/4)*exp(3*x)*exp(x^(2*x)/x^4)*exp(9)*exp(x^x/x)*exp((6*x^x) /x^2)
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.50 \[ \int \frac {2 x^5+e^{\frac {36 x^4+12 x^5+x^6+4 x^{2 x}+x^x \left (24 x^2+4 x^3\right )}{4 x^4}} \left (6 x^5+x^6+x^{2 x} (-8+4 x+4 x \log (x))+x^x \left (-24 x^2+10 x^3+2 x^4+\left (12 x^3+2 x^4\right ) \log (x)\right )\right )}{2 x^5} \, dx=e^{\frac {4 x^{2 x}+4 x^{x} x^{3}+24 x^{x} x^{2}+x^{6}+12 x^{5}}{4 x^{4}}} e^{9}+x \] Input:
int(1/2*(((4*x*log(x)+4*x-8)*exp(x*log(x))^2+((2*x^4+12*x^3)*log(x)+2*x^4+ 10*x^3-24*x^2)*exp(x*log(x))+x^6+6*x^5)*exp(1/4*(4*exp(x*log(x))^2+(4*x^3+ 24*x^2)*exp(x*log(x))+x^6+12*x^5+36*x^4)/x^4)+2*x^5)/x^5,x)
Output:
e**((4*x**(2*x) + 4*x**x*x**3 + 24*x**x*x**2 + x**6 + 12*x**5)/(4*x**4))*e **9 + x