Integrand size = 61, antiderivative size = 24 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=e^{\frac {e^{5-x-\frac {5}{4} x (4+\log (x))}}{x}}+x \] Output:
exp(exp(5-5*x*(1/4*ln(x)+1)-x)/x)+x
Time = 0.50 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=e^{e^{5-6 x} x^{-1-\frac {5 x}{4}}}+x \] Input:
Integrate[(4*x^2 + E^(E^((20 - 24*x - 5*x*Log[x])/4)/x + (20 - 24*x - 5*x* Log[x])/4)*(-4 - 29*x - 5*x*Log[x]))/(4*x^2),x]
Output:
E^(E^(5 - 6*x)*x^(-1 - (5*x)/4)) + 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.
\(\displaystyle \int \frac {(-29 x-5 x \log (x)-4) \exp \left (\frac {1}{4} (-24 x-5 x \log (x)+20)+\frac {e^{\frac {1}{4} (-24 x-5 x \log (x)+20)}}{x}\right )+4 x^2}{4 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {4 x^2-\exp \left (e^{5-6 x} x^{-\frac {5 x}{4}-1}+\frac {1}{4} (-5 \log (x) x-24 x+20)\right ) (5 \log (x) x+29 x+4)}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {1}{4} \int \left (4-e^{e^{5-6 x} x^{-\frac {5 x}{4}-1}-6 x+5} x^{-\frac {5 x}{4}-2} (5 \log (x) x+29 x+4)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-4 \int e^{e^{5-6 x} x^{-\frac {5 x}{4}-1}-6 x+5} x^{-\frac {5 x}{4}-2}dx-29 \int e^{e^{5-6 x} x^{-\frac {5 x}{4}-1}-6 x+5} x^{-\frac {5 x}{4}-1}dx+5 \int \frac {\int e^{e^{5-6 x} x^{-\frac {5 x}{4}-1}-6 x+5} x^{-\frac {5 x}{4}-1}dx}{x}dx-5 \log (x) \int e^{e^{5-6 x} x^{-\frac {5 x}{4}-1}-6 x+5} x^{-\frac {5 x}{4}-1}dx+4 x\right )\) |
Input:
Int[(4*x^2 + E^(E^((20 - 24*x - 5*x*Log[x])/4)/x + (20 - 24*x - 5*x*Log[x] )/4)*(-4 - 29*x - 5*x*Log[x]))/(4*x^2),x]
Output:
$Aborted
Time = 0.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79
method | result | size |
risch | \(x +{\mathrm e}^{\frac {x^{-\frac {5 x}{4}} {\mathrm e}^{5-6 x}}{x}}\) | \(19\) |
parallelrisch | \(x +{\mathrm e}^{\frac {{\mathrm e}^{-\frac {5 x \ln \left (x \right )}{4}-6 x +5}}{x}}\) | \(19\) |
Input:
int(1/4*((-5*x*ln(x)-29*x-4)*exp(-5/4*x*ln(x)-6*x+5)*exp(exp(-5/4*x*ln(x)- 6*x+5)/x)+4*x^2)/x^2,x,method=_RETURNVERBOSE)
Output:
x+exp(x^(-5/4*x)*exp(5-6*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx={\left (x e^{\left (-\frac {5}{4} \, x \log \left (x\right ) - 6 \, x + 5\right )} + e^{\left (-\frac {5 \, x^{2} \log \left (x\right ) + 24 \, x^{2} - 20 \, x - 4 \, e^{\left (-\frac {5}{4} \, x \log \left (x\right ) - 6 \, x + 5\right )}}{4 \, x}\right )}\right )} e^{\left (\frac {5}{4} \, x \log \left (x\right ) + 6 \, x - 5\right )} \] Input:
integrate(1/4*((-5*x*log(x)-29*x-4)*exp(-5/4*x*log(x)-6*x+5)*exp(exp(-5/4* x*log(x)-6*x+5)/x)+4*x^2)/x^2,x, algorithm="fricas")
Output:
(x*e^(-5/4*x*log(x) - 6*x + 5) + e^(-1/4*(5*x^2*log(x) + 24*x^2 - 20*x - 4 *e^(-5/4*x*log(x) - 6*x + 5))/x))*e^(5/4*x*log(x) + 6*x - 5)
Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=x + e^{\frac {e^{- \frac {5 x \log {\left (x \right )}}{4} - 6 x + 5}}{x}} \] Input:
integrate(1/4*((-5*x*ln(x)-29*x-4)*exp(-5/4*x*ln(x)-6*x+5)*exp(exp(-5/4*x* ln(x)-6*x+5)/x)+4*x**2)/x**2,x)
Output:
x + exp(exp(-5*x*log(x)/4 - 6*x + 5)/x)
Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=x + e^{\left (\frac {e^{\left (-\frac {5}{4} \, x \log \left (x\right ) - 6 \, x + 5\right )}}{x}\right )} \] Input:
integrate(1/4*((-5*x*log(x)-29*x-4)*exp(-5/4*x*log(x)-6*x+5)*exp(exp(-5/4* x*log(x)-6*x+5)/x)+4*x^2)/x^2,x, algorithm="maxima")
Output:
x + e^(e^(-5/4*x*log(x) - 6*x + 5)/x)
\[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=\int { \frac {4 \, x^{2} - {\left (5 \, x \log \left (x\right ) + 29 \, x + 4\right )} e^{\left (-\frac {5}{4} \, x \log \left (x\right ) - 6 \, x + \frac {e^{\left (-\frac {5}{4} \, x \log \left (x\right ) - 6 \, x + 5\right )}}{x} + 5\right )}}{4 \, x^{2}} \,d x } \] Input:
integrate(1/4*((-5*x*log(x)-29*x-4)*exp(-5/4*x*log(x)-6*x+5)*exp(exp(-5/4* x*log(x)-6*x+5)/x)+4*x^2)/x^2,x, algorithm="giac")
Output:
integrate(1/4*(4*x^2 - (5*x*log(x) + 29*x + 4)*e^(-5/4*x*log(x) - 6*x + e^ (-5/4*x*log(x) - 6*x + 5)/x + 5))/x^2, x)
Time = 3.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=x+{\mathrm {e}}^{\frac {{\mathrm {e}}^{-6\,x}\,{\mathrm {e}}^5}{x^{\frac {5\,x}{4}}\,x}} \] Input:
int((x^2 - (exp(5 - (5*x*log(x))/4 - 6*x)*exp(exp(5 - (5*x*log(x))/4 - 6*x )/x)*(29*x + 5*x*log(x) + 4))/4)/x^2,x)
Output:
x + exp((exp(-6*x)*exp(5))/(x^((5*x)/4)*x))
\[ \int \frac {4 x^2+e^{\frac {e^{\frac {1}{4} (20-24 x-5 x \log (x))}}{x}+\frac {1}{4} (20-24 x-5 x \log (x))} (-4-29 x-5 x \log (x))}{4 x^2} \, dx=-\left (\int \frac {e^{\frac {e^{5}}{x^{\frac {5 x}{4}} e^{6 x} x}}}{x^{\frac {5 x}{4}} e^{6 x} x^{2}}d x \right ) e^{5}-\frac {29 \left (\int \frac {e^{\frac {e^{5}}{x^{\frac {5 x}{4}} e^{6 x} x}}}{x^{\frac {5 x}{4}} e^{6 x} x}d x \right ) e^{5}}{4}-\frac {5 \left (\int \frac {e^{\frac {e^{5}}{x^{\frac {5 x}{4}} e^{6 x} x}} \mathrm {log}\left (x \right )}{x^{\frac {5 x}{4}} e^{6 x} x}d x \right ) e^{5}}{4}+x \] Input:
int(1/4*((-5*x*log(x)-29*x-4)*exp(-5/4*x*log(x)-6*x+5)*exp(exp(-5/4*x*log( x)-6*x+5)/x)+4*x^2)/x^2,x)
Output:
( - 4*int(e**(e**5/(x**((5*x)/4)*e**(6*x)*x))/(x**((5*x)/4)*e**(6*x)*x**2) ,x)*e**5 - 29*int(e**(e**5/(x**((5*x)/4)*e**(6*x)*x))/(x**((5*x)/4)*e**(6* x)*x),x)*e**5 - 5*int((e**(e**5/(x**((5*x)/4)*e**(6*x)*x))*log(x))/(x**((5 *x)/4)*e**(6*x)*x),x)*e**5 + 4*x)/4