Integrand size = 274, antiderivative size = 27 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2}{x+\frac {x^2}{5+x}+\log \left (1+e^3+x+x^2+\log (x)\right )} \] Output:
2/(x^2/(5+x)+x+ln(ln(x)+exp(3)+x^2+x+1))
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 (5+x)}{x (5+2 x)+(5+x) \log \left (1+e^3+x+x^2+\log (x)\right )} \] Input:
Integrate[(-50 - 120*x - 212*x^2 - 136*x^3 - 48*x^4 - 4*x^5 + E^3*(-50*x - 40*x^2 - 4*x^3) + (-50*x - 40*x^2 - 4*x^3)*Log[x])/(25*x^3 + 45*x^4 + 49* x^5 + 24*x^6 + 4*x^7 + E^3*(25*x^3 + 20*x^4 + 4*x^5) + (25*x^3 + 20*x^4 + 4*x^5)*Log[x] + (50*x^2 + 80*x^3 + 84*x^4 + 34*x^5 + 4*x^6 + E^3*(50*x^2 + 30*x^3 + 4*x^4) + (50*x^2 + 30*x^3 + 4*x^4)*Log[x])*Log[1 + E^3 + x + x^2 + Log[x]] + (25*x + 35*x^2 + 36*x^3 + 11*x^4 + x^5 + E^3*(25*x + 10*x^2 + x^3) + (25*x + 10*x^2 + x^3)*Log[x])*Log[1 + E^3 + x + x^2 + Log[x]]^2),x ]
Output:
(2*(5 + x))/(x*(5 + 2*x) + (5 + x)*Log[1 + E^3 + x + x^2 + Log[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.
| \(\displaystyle \int \frac {-4 x^5-48 x^4-136 x^3-212 x^2+e^3 \left (-4 x^3-40 x^2-50 x\right )+\left (-4 x^3-40 x^2-50 x\right ) \log (x)-120 x-50}{4 x^7+24 x^6+49 x^5+45 x^4+25 x^3+e^3 \left (4 x^5+20 x^4+25 x^3\right )+\left (4 x^5+20 x^4+25 x^3\right ) \log (x)+\left (x^5+11 x^4+36 x^3+35 x^2+e^3 \left (x^3+10 x^2+25 x\right )+\left (x^3+10 x^2+25 x\right ) \log (x)+25 x\right ) \log ^2\left (x^2+x+\log (x)+e^3+1\right )+\left (4 x^6+34 x^5+84 x^4+80 x^3+50 x^2+e^3 \left (4 x^4+30 x^3+50 x^2\right )+\left (4 x^4+30 x^3+50 x^2\right ) \log (x)\right ) \log \left (x^2+x+\log (x)+e^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-2 x^5-24 x^4-2 \left (34+e^3\right ) x^3-2 \left (53+10 e^3\right ) x^2-\left (2 x^2+20 x+25\right ) x \log (x)-5 \left (12+5 e^3\right ) x-25\right )}{x \left (x^2+x+\log (x)+e^3+1\right ) \left ((x+5) \log \left (x^2+x+\log (x)+e^3+1\right )+x (2 x+5)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {2 x^5+24 x^4+2 \left (34+e^3\right ) x^3+2 \left (53+10 e^3\right ) x^2+\left (2 x^2+20 x+25\right ) \log (x) x+5 \left (12+5 e^3\right ) x+25}{x \left (x^2+x+\log (x)+e^3+1\right ) \left (x (2 x+5)+(x+5) \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {2 x^5+24 x^4+2 \left (34+e^3\right ) x^3+2 \left (53+10 e^3\right ) x^2+\left (2 x^2+20 x+25\right ) \log (x) x+5 \left (12+5 e^3\right ) x+25}{x \left (x^2+x+\log (x)+e^3+1\right ) \left (x (2 x+5)+(x+5) \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {2 x^4}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {24 x^3}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {2 \log (x) x^2}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {2 \left (34+e^3\right ) x^2}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {20 \log (x) x}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {2 \left (53+10 e^3\right ) x}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {25 \log (x)}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {5 \left (12+5 e^3\right )}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}+\frac {25}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (5 \left (12+5 e^3\right ) \int \frac {1}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+25 \int \frac {1}{x \left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+2 \left (53+10 e^3\right ) \int \frac {x}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+2 \left (34+e^3\right ) \int \frac {x^2}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+25 \int \frac {\log (x)}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+20 \int \frac {x \log (x)}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+2 \int \frac {x^2 \log (x)}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+2 \int \frac {x^4}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx+24 \int \frac {x^3}{\left (x^2+x+\log (x)+e^3+1\right ) \left (2 x^2+\log \left (x^2+x+\log (x)+e^3+1\right ) x+5 x+5 \log \left (x^2+x+\log (x)+e^3+1\right )\right )^2}dx\right )\) |
Input:
Int[(-50 - 120*x - 212*x^2 - 136*x^3 - 48*x^4 - 4*x^5 + E^3*(-50*x - 40*x^ 2 - 4*x^3) + (-50*x - 40*x^2 - 4*x^3)*Log[x])/(25*x^3 + 45*x^4 + 49*x^5 + 24*x^6 + 4*x^7 + E^3*(25*x^3 + 20*x^4 + 4*x^5) + (25*x^3 + 20*x^4 + 4*x^5) *Log[x] + (50*x^2 + 80*x^3 + 84*x^4 + 34*x^5 + 4*x^6 + E^3*(50*x^2 + 30*x^ 3 + 4*x^4) + (50*x^2 + 30*x^3 + 4*x^4)*Log[x])*Log[1 + E^3 + x + x^2 + Log [x]] + (25*x + 35*x^2 + 36*x^3 + 11*x^4 + x^5 + E^3*(25*x + 10*x^2 + x^3) + (25*x + 10*x^2 + x^3)*Log[x])*Log[1 + E^3 + x + x^2 + Log[x]]^2),x]
Output:
$Aborted
Time = 15.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59
| method | result | size |
| default | \(\frac {2 x +10}{x \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+2 x^{2}+5 \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+5 x}\) | \(43\) |
| risch | \(\frac {2 x +10}{x \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+2 x^{2}+5 \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+5 x}\) | \(43\) |
| parallelrisch | \(\frac {2 x +10}{x \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+2 x^{2}+5 \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+x^{2}+x +1\right )+5 x}\) | \(44\) |
Input:
int(((-4*x^3-40*x^2-50*x)*ln(x)+(-4*x^3-40*x^2-50*x)*exp(3)-4*x^5-48*x^4-1 36*x^3-212*x^2-120*x-50)/(((x^3+10*x^2+25*x)*ln(x)+(x^3+10*x^2+25*x)*exp(3 )+x^5+11*x^4+36*x^3+35*x^2+25*x)*ln(ln(x)+exp(3)+x^2+x+1)^2+((4*x^4+30*x^3 +50*x^2)*ln(x)+(4*x^4+30*x^3+50*x^2)*exp(3)+4*x^6+34*x^5+84*x^4+80*x^3+50* x^2)*ln(ln(x)+exp(3)+x^2+x+1)+(4*x^5+20*x^4+25*x^3)*ln(x)+(4*x^5+20*x^4+25 *x^3)*exp(3)+4*x^7+24*x^6+49*x^5+45*x^4+25*x^3),x,method=_RETURNVERBOSE)
Output:
2*(5+x)/(x*ln(ln(x)+exp(3)+x^2+x+1)+2*x^2+5*ln(ln(x)+exp(3)+x^2+x+1)+5*x)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + {\left (x + 5\right )} \log \left (x^{2} + x + e^{3} + \log \left (x\right ) + 1\right ) + 5 \, x} \] Input:
integrate(((-4*x^3-40*x^2-50*x)*log(x)+(-4*x^3-40*x^2-50*x)*exp(3)-4*x^5-4 8*x^4-136*x^3-212*x^2-120*x-50)/(((x^3+10*x^2+25*x)*log(x)+(x^3+10*x^2+25* x)*exp(3)+x^5+11*x^4+36*x^3+35*x^2+25*x)*log(log(x)+exp(3)+x^2+x+1)^2+((4* x^4+30*x^3+50*x^2)*log(x)+(4*x^4+30*x^3+50*x^2)*exp(3)+4*x^6+34*x^5+84*x^4 +80*x^3+50*x^2)*log(log(x)+exp(3)+x^2+x+1)+(4*x^5+20*x^4+25*x^3)*log(x)+(4 *x^5+20*x^4+25*x^3)*exp(3)+4*x^7+24*x^6+49*x^5+45*x^4+25*x^3),x, algorithm ="fricas")
Output:
2*(x + 5)/(2*x^2 + (x + 5)*log(x^2 + x + e^3 + log(x) + 1) + 5*x)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 x + 10}{2 x^{2} + 5 x + \left (x + 5\right ) \log {\left (x^{2} + x + \log {\left (x \right )} + 1 + e^{3} \right )}} \] Input:
integrate(((-4*x**3-40*x**2-50*x)*ln(x)+(-4*x**3-40*x**2-50*x)*exp(3)-4*x* *5-48*x**4-136*x**3-212*x**2-120*x-50)/(((x**3+10*x**2+25*x)*ln(x)+(x**3+1 0*x**2+25*x)*exp(3)+x**5+11*x**4+36*x**3+35*x**2+25*x)*ln(ln(x)+exp(3)+x** 2+x+1)**2+((4*x**4+30*x**3+50*x**2)*ln(x)+(4*x**4+30*x**3+50*x**2)*exp(3)+ 4*x**6+34*x**5+84*x**4+80*x**3+50*x**2)*ln(ln(x)+exp(3)+x**2+x+1)+(4*x**5+ 20*x**4+25*x**3)*ln(x)+(4*x**5+20*x**4+25*x**3)*exp(3)+4*x**7+24*x**6+49*x **5+45*x**4+25*x**3),x)
Output:
(2*x + 10)/(2*x**2 + 5*x + (x + 5)*log(x**2 + x + log(x) + 1 + exp(3)))
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + {\left (x + 5\right )} \log \left (x^{2} + x + e^{3} + \log \left (x\right ) + 1\right ) + 5 \, x} \] Input:
integrate(((-4*x^3-40*x^2-50*x)*log(x)+(-4*x^3-40*x^2-50*x)*exp(3)-4*x^5-4 8*x^4-136*x^3-212*x^2-120*x-50)/(((x^3+10*x^2+25*x)*log(x)+(x^3+10*x^2+25* x)*exp(3)+x^5+11*x^4+36*x^3+35*x^2+25*x)*log(log(x)+exp(3)+x^2+x+1)^2+((4* x^4+30*x^3+50*x^2)*log(x)+(4*x^4+30*x^3+50*x^2)*exp(3)+4*x^6+34*x^5+84*x^4 +80*x^3+50*x^2)*log(log(x)+exp(3)+x^2+x+1)+(4*x^5+20*x^4+25*x^3)*log(x)+(4 *x^5+20*x^4+25*x^3)*exp(3)+4*x^7+24*x^6+49*x^5+45*x^4+25*x^3),x, algorithm ="maxima")
Output:
2*(x + 5)/(2*x^2 + (x + 5)*log(x^2 + x + e^3 + log(x) + 1) + 5*x)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 \, {\left (x + 5\right )}}{2 \, x^{2} + x \log \left (x^{2} + x + e^{3} + \log \left (x\right ) + 1\right ) + 5 \, x + 5 \, \log \left (x^{2} + x + e^{3} + \log \left (x\right ) + 1\right )} \] Input:
integrate(((-4*x^3-40*x^2-50*x)*log(x)+(-4*x^3-40*x^2-50*x)*exp(3)-4*x^5-4 8*x^4-136*x^3-212*x^2-120*x-50)/(((x^3+10*x^2+25*x)*log(x)+(x^3+10*x^2+25* x)*exp(3)+x^5+11*x^4+36*x^3+35*x^2+25*x)*log(log(x)+exp(3)+x^2+x+1)^2+((4* x^4+30*x^3+50*x^2)*log(x)+(4*x^4+30*x^3+50*x^2)*exp(3)+4*x^6+34*x^5+84*x^4 +80*x^3+50*x^2)*log(log(x)+exp(3)+x^2+x+1)+(4*x^5+20*x^4+25*x^3)*log(x)+(4 *x^5+20*x^4+25*x^3)*exp(3)+4*x^7+24*x^6+49*x^5+45*x^4+25*x^3),x, algorithm ="giac")
Output:
2*(x + 5)/(2*x^2 + x*log(x^2 + x + e^3 + log(x) + 1) + 5*x + 5*log(x^2 + x + e^3 + log(x) + 1))
Timed out. \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\int -\frac {120\,x+{\mathrm {e}}^3\,\left (4\,x^3+40\,x^2+50\,x\right )+212\,x^2+136\,x^3+48\,x^4+4\,x^5+\ln \left (x\right )\,\left (4\,x^3+40\,x^2+50\,x\right )+50}{\ln \left (x+{\mathrm {e}}^3+\ln \left (x\right )+x^2+1\right )\,\left (\ln \left (x\right )\,\left (4\,x^4+30\,x^3+50\,x^2\right )+{\mathrm {e}}^3\,\left (4\,x^4+30\,x^3+50\,x^2\right )+50\,x^2+80\,x^3+84\,x^4+34\,x^5+4\,x^6\right )+\ln \left (x\right )\,\left (4\,x^5+20\,x^4+25\,x^3\right )+{\ln \left (x+{\mathrm {e}}^3+\ln \left (x\right )+x^2+1\right )}^2\,\left (25\,x+{\mathrm {e}}^3\,\left (x^3+10\,x^2+25\,x\right )+\ln \left (x\right )\,\left (x^3+10\,x^2+25\,x\right )+35\,x^2+36\,x^3+11\,x^4+x^5\right )+{\mathrm {e}}^3\,\left (4\,x^5+20\,x^4+25\,x^3\right )+25\,x^3+45\,x^4+49\,x^5+24\,x^6+4\,x^7} \,d x \] Input:
int(-(120*x + exp(3)*(50*x + 40*x^2 + 4*x^3) + 212*x^2 + 136*x^3 + 48*x^4 + 4*x^5 + log(x)*(50*x + 40*x^2 + 4*x^3) + 50)/(log(x + exp(3) + log(x) + x^2 + 1)*(log(x)*(50*x^2 + 30*x^3 + 4*x^4) + exp(3)*(50*x^2 + 30*x^3 + 4*x ^4) + 50*x^2 + 80*x^3 + 84*x^4 + 34*x^5 + 4*x^6) + log(x)*(25*x^3 + 20*x^4 + 4*x^5) + log(x + exp(3) + log(x) + x^2 + 1)^2*(25*x + exp(3)*(25*x + 10 *x^2 + x^3) + log(x)*(25*x + 10*x^2 + x^3) + 35*x^2 + 36*x^3 + 11*x^4 + x^ 5) + exp(3)*(25*x^3 + 20*x^4 + 4*x^5) + 25*x^3 + 45*x^4 + 49*x^5 + 24*x^6 + 4*x^7),x)
Output:
int(-(120*x + exp(3)*(50*x + 40*x^2 + 4*x^3) + 212*x^2 + 136*x^3 + 48*x^4 + 4*x^5 + log(x)*(50*x + 40*x^2 + 4*x^3) + 50)/(log(x + exp(3) + log(x) + x^2 + 1)*(log(x)*(50*x^2 + 30*x^3 + 4*x^4) + exp(3)*(50*x^2 + 30*x^3 + 4*x ^4) + 50*x^2 + 80*x^3 + 84*x^4 + 34*x^5 + 4*x^6) + log(x)*(25*x^3 + 20*x^4 + 4*x^5) + log(x + exp(3) + log(x) + x^2 + 1)^2*(25*x + exp(3)*(25*x + 10 *x^2 + x^3) + log(x)*(25*x + 10*x^2 + x^3) + 35*x^2 + 36*x^3 + 11*x^4 + x^ 5) + exp(3)*(25*x^3 + 20*x^4 + 4*x^5) + 25*x^3 + 45*x^4 + 49*x^5 + 24*x^6 + 4*x^7), x)
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-50-120 x-212 x^2-136 x^3-48 x^4-4 x^5+e^3 \left (-50 x-40 x^2-4 x^3\right )+\left (-50 x-40 x^2-4 x^3\right ) \log (x)}{25 x^3+45 x^4+49 x^5+24 x^6+4 x^7+e^3 \left (25 x^3+20 x^4+4 x^5\right )+\left (25 x^3+20 x^4+4 x^5\right ) \log (x)+\left (50 x^2+80 x^3+84 x^4+34 x^5+4 x^6+e^3 \left (50 x^2+30 x^3+4 x^4\right )+\left (50 x^2+30 x^3+4 x^4\right ) \log (x)\right ) \log \left (1+e^3+x+x^2+\log (x)\right )+\left (25 x+35 x^2+36 x^3+11 x^4+x^5+e^3 \left (25 x+10 x^2+x^3\right )+\left (25 x+10 x^2+x^3\right ) \log (x)\right ) \log ^2\left (1+e^3+x+x^2+\log (x)\right )} \, dx=\frac {2 x +10}{\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{3}+x^{2}+x +1\right ) x +5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )+e^{3}+x^{2}+x +1\right )+2 x^{2}+5 x} \] Input:
int(((-4*x^3-40*x^2-50*x)*log(x)+(-4*x^3-40*x^2-50*x)*exp(3)-4*x^5-48*x^4- 136*x^3-212*x^2-120*x-50)/(((x^3+10*x^2+25*x)*log(x)+(x^3+10*x^2+25*x)*exp (3)+x^5+11*x^4+36*x^3+35*x^2+25*x)*log(log(x)+exp(3)+x^2+x+1)^2+((4*x^4+30 *x^3+50*x^2)*log(x)+(4*x^4+30*x^3+50*x^2)*exp(3)+4*x^6+34*x^5+84*x^4+80*x^ 3+50*x^2)*log(log(x)+exp(3)+x^2+x+1)+(4*x^5+20*x^4+25*x^3)*log(x)+(4*x^5+2 0*x^4+25*x^3)*exp(3)+4*x^7+24*x^6+49*x^5+45*x^4+25*x^3),x)
Output:
(2*(x + 5))/(log(log(x) + e**3 + x**2 + x + 1)*x + 5*log(log(x) + e**3 + x **2 + x + 1) + 2*x**2 + 5*x)