Integrand size = 154, antiderivative size = 28 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=\frac {1}{x}+x-\frac {25}{\log ^2\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \]
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=\frac {1}{x}+x-\frac {25}{\log ^2\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \]
Integrate[(50*x - 50*x^2 + E^(4*x + x^2)*(-200*x^2 - 100*x^3) + (x - x^3 + E^(4*x + x^2)*(1 - x^2) + E^3*(-1 + x^2) + (-1 + x^2)*Log[x])*Log[-E^3 + E^(4*x + x^2) + x - Log[x]]^3)/((E^3*x^2 - E^(4*x + x^2)*x^2 - x^3 + x^2*L og[x])*Log[-E^3 + E^(4*x + x^2) + x - Log[x]]^3),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.
\(\displaystyle \int \frac {-50 x^2+e^{x^2+4 x} \left (-100 x^3-200 x^2\right )+\left (-x^3+e^{x^2+4 x} \left (1-x^2\right )+e^3 \left (x^2-1\right )+\left (x^2-1\right ) \log (x)+x\right ) \log ^3\left (e^{x^2+4 x}+x-\log (x)-e^3\right )+50 x}{\left (-x^3-e^{x^2+4 x} x^2+e^3 x^2+x^2 \log (x)\right ) \log ^3\left (e^{x^2+4 x}+x-\log (x)-e^3\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {50 \left (2 x^3+4 \left (1-\frac {e^3}{2}\right ) x^2-2 x^2 \log (x)-\left (1+4 e^3\right ) x-4 x \log (x)+1\right )}{x \left (-x-e^{x (x+4)}+\log (x)+e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}+\frac {100 x^3+200 x^2+x^2 \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )-\log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}{x^2 \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -100 \int \frac {x^2}{\left (x+e^{x (x+4)}-\log (x)-e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx+200 \int \frac {1}{\log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx+100 \int \frac {x}{\log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx-50 \int \frac {1}{x \left (x+e^{x (x+4)}-\log (x)-e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx+100 \int \frac {x \log (x)}{\left (x+e^{x (x+4)}-\log (x)-e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx-50 \left (1+4 e^3\right ) \int \frac {1}{\left (-x-e^{x (x+4)}+\log (x)+e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx+100 \left (2-e^3\right ) \int \frac {x}{\left (-x-e^{x (x+4)}+\log (x)+e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx-200 \int \frac {\log (x)}{\left (-x-e^{x (x+4)}+\log (x)+e^3\right ) \log ^3\left (x+e^{x (x+4)}-\log (x)-e^3\right )}dx+x+\frac {1}{x}\) |
Int[(50*x - 50*x^2 + E^(4*x + x^2)*(-200*x^2 - 100*x^3) + (x - x^3 + E^(4* x + x^2)*(1 - x^2) + E^3*(-1 + x^2) + (-1 + x^2)*Log[x])*Log[-E^3 + E^(4*x + x^2) + x - Log[x]]^3)/((E^3*x^2 - E^(4*x + x^2)*x^2 - x^3 + x^2*Log[x]) *Log[-E^3 + E^(4*x + x^2) + x - Log[x]]^3),x]
3.13.52.3.1 Defintions of rubi rules used
Time = 11.72 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {x^{2}+1}{x}-\frac {25}{\ln \left (x +{\mathrm e}^{\left (4+x \right ) x}-{\mathrm e}^{3}-\ln \left (x \right )\right )^{2}}\) | \(32\) |
parallelrisch | \(\frac {\ln \left (-\ln \left (x \right )+{\mathrm e}^{x^{2}+4 x}+x -{\mathrm e}^{3}\right )^{2} x^{2}+\ln \left (-\ln \left (x \right )+{\mathrm e}^{x^{2}+4 x}+x -{\mathrm e}^{3}\right )^{2}-25 x}{x \ln \left (-\ln \left (x \right )+{\mathrm e}^{x^{2}+4 x}+x -{\mathrm e}^{3}\right )^{2}}\) | \(76\) |
int((((x^2-1)*ln(x)+(-x^2+1)*exp(x^2+4*x)+(x^2-1)*exp(3)-x^3+x)*ln(-ln(x)+ exp(x^2+4*x)+x-exp(3))^3+(-100*x^3-200*x^2)*exp(x^2+4*x)-50*x^2+50*x)/(x^2 *ln(x)-x^2*exp(x^2+4*x)+x^2*exp(3)-x^3)/ln(-ln(x)+exp(x^2+4*x)+x-exp(3))^3 ,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=\frac {{\left (x^{2} + 1\right )} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2}} \]
integrate((((x^2-1)*log(x)+(-x^2+1)*exp(x^2+4*x)+(x^2-1)*exp(3)-x^3+x)*log (-log(x)+exp(x^2+4*x)+x-exp(3))^3+(-100*x^3-200*x^2)*exp(x^2+4*x)-50*x^2+5 0*x)/(x^2*log(x)-x^2*exp(x^2+4*x)+x^2*exp(3)-x^3)/log(-log(x)+exp(x^2+4*x) +x-exp(3))^3,x, algorithm=\
((x^2 + 1)*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2 - 25*x)/(x*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2)
Time = 1.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=x - \frac {25}{\log {\left (x + e^{x^{2} + 4 x} - \log {\left (x \right )} - e^{3} \right )}^{2}} + \frac {1}{x} \]
integrate((((x**2-1)*ln(x)+(-x**2+1)*exp(x**2+4*x)+(x**2-1)*exp(3)-x**3+x) *ln(-ln(x)+exp(x**2+4*x)+x-exp(3))**3+(-100*x**3-200*x**2)*exp(x**2+4*x)-5 0*x**2+50*x)/(x**2*ln(x)-x**2*exp(x**2+4*x)+x**2*exp(3)-x**3)/ln(-ln(x)+ex p(x**2+4*x)+x-exp(3))**3,x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=\frac {{\left (x^{2} + 1\right )} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2}} \]
integrate((((x^2-1)*log(x)+(-x^2+1)*exp(x^2+4*x)+(x^2-1)*exp(3)-x^3+x)*log (-log(x)+exp(x^2+4*x)+x-exp(3))^3+(-100*x^3-200*x^2)*exp(x^2+4*x)-50*x^2+5 0*x)/(x^2*log(x)-x^2*exp(x^2+4*x)+x^2*exp(3)-x^3)/log(-log(x)+exp(x^2+4*x) +x-exp(3))^3,x, algorithm=\
((x^2 + 1)*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2 - 25*x)/(x*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 1.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=\frac {x^{2} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2} + \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \left (x\right )\right )^{2}} \]
integrate((((x^2-1)*log(x)+(-x^2+1)*exp(x^2+4*x)+(x^2-1)*exp(3)-x^3+x)*log (-log(x)+exp(x^2+4*x)+x-exp(3))^3+(-100*x^3-200*x^2)*exp(x^2+4*x)-50*x^2+5 0*x)/(x^2*log(x)-x^2*exp(x^2+4*x)+x^2*exp(3)-x^3)/log(-log(x)+exp(x^2+4*x) +x-exp(3))^3,x, algorithm=\
(x^2*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2 + log(x - e^3 + e^(x^2 + 4*x) - log(x))^2 - 25*x)/(x*log(x - e^3 + e^(x^2 + 4*x) - log(x))^2)
Time = 8.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx=x-\frac {25}{{\ln \left (x-{\mathrm {e}}^3-\ln \left (x\right )+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^2}\right )}^2}+\frac {1}{x} \]