Integrand size = 170, antiderivative size = 28 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=e^4 \left (9+x \left (5-x^2-\log (x+\log (-4+x-\log (x)))\right )\right ) \]
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=e^4 \left (5 x-x^3-x \log (x+\log (-4+x-\log (x)))\right ) \]
Integrate[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Lo g[x] + (E^4*(20 - 5*x - 12*x^2 + 3*x^3) + E^4*(5 - 3*x^2)*Log[x])*Log[-4 + x - Log[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Log[ x])*Log[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Lo g[x] + (4 - x + Log[x])*Log[-4 + x - Log[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.
\(\displaystyle \int \frac {e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (x^2-4 x\right )-e^4 x \log (x)+\left (e^4 (x-4)-e^4 \log (x)\right ) \log (x-\log (x)-4)\right ) \log (x+\log (x-\log (x)-4))+\left (e^4 \left (5-3 x^2\right ) \log (x)+e^4 \left (3 x^3-12 x^2-5 x+20\right )\right ) \log (x-\log (x)-4)+e^4 \left (3 x^4-12 x^3-4 x^2+17 x-1\right )}{-x^2+4 x+x \log (x)+(-x+\log (x)+4) \log (x-\log (x)-4)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (x^2-4 x\right )-e^4 x \log (x)+\left (e^4 (x-4)-e^4 \log (x)\right ) \log (x-\log (x)-4)\right ) \log (x+\log (x-\log (x)-4))+\left (e^4 \left (5-3 x^2\right ) \log (x)+e^4 \left (3 x^3-12 x^2-5 x+20\right )\right ) \log (x-\log (x)-4)+e^4 \left (3 x^4-12 x^3-4 x^2+17 x-1\right )}{(-x+\log (x)+4) (x+\log (x-\log (x)-4))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^4 \left (3 x^4-12 x^3-3 x^3 \log (x)+3 x^3 \log (x-\log (x)-4)-4 x^2-3 x^2 \log (x) \log (x-\log (x)-4)-12 x^2 \log (x-\log (x)-4)+17 x+4 x \log (x)-5 x \log (x-\log (x)-4)+5 \log (x) \log (x-\log (x)-4)+20 \log (x-\log (x)-4)-1\right )}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}-e^4 \log (x+\log (x-\log (x)-4))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^4 \int \frac {x^2}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+e^4 \int \frac {1}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+3 e^4 \int \frac {x}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx+e^4 \int \frac {x \log (x)}{(x-\log (x)-4) (x+\log (x-\log (x)-4))}dx-e^4 \int \log (x+\log (x-\log (x)-4))dx-e^4 x^3+5 e^4 x\) |
Int[(E^4*(-1 + 17*x - 4*x^2 - 12*x^3 + 3*x^4) + E^4*(4*x - 3*x^3)*Log[x] + (E^4*(20 - 5*x - 12*x^2 + 3*x^3) + E^4*(5 - 3*x^2)*Log[x])*Log[-4 + x - L og[x]] + (E^4*(-4*x + x^2) - E^4*x*Log[x] + (E^4*(-4 + x) - E^4*Log[x])*Lo g[-4 + x - Log[x]])*Log[x + Log[-4 + x - Log[x]]])/(4*x - x^2 + x*Log[x] + (4 - x + Log[x])*Log[-4 + x - Log[x]]),x]
3.11.3.3.1 Defintions of rubi rules used
Time = 6.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-x \,{\mathrm e}^{4} \ln \left (\ln \left (-\ln \left (x \right )+x -4\right )+x \right )-{\mathrm e}^{4} x \left (x^{2}-5\right )\) | \(28\) |
parallelrisch | \(-x^{3} {\mathrm e}^{4}-x \,{\mathrm e}^{4} \ln \left (\ln \left (-\ln \left (x \right )+x -4\right )+x \right )+5 x \,{\mathrm e}^{4}\) | \(30\) |
int((((-exp(4)*ln(x)+(x-4)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x^2-4*x) *exp(4))*ln(ln(-ln(x)+x-4)+x)+((-3*x^2+5)*exp(4)*ln(x)+(3*x^3-12*x^2-5*x+2 0)*exp(4))*ln(-ln(x)+x-4)+(-3*x^3+4*x)*exp(4)*ln(x)+(3*x^4-12*x^3-4*x^2+17 *x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x^2+4*x),x,method=_RETUR NVERBOSE)
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) - {\left (x^{3} - 5 \, x\right )} e^{4} \]
integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x) +(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^ 3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x^ 4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2 +4*x),x, algorithm=\
Exception generated. \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=\text {Exception raised: TypeError} \]
integrate((((-exp(4)*ln(x)+(x-4)*exp(4))*ln(-ln(x)+x-4)-x*exp(4)*ln(x)+(x* *2-4*x)*exp(4))*ln(ln(-ln(x)+x-4)+x)+((-3*x**2+5)*exp(4)*ln(x)+(3*x**3-12* x**2-5*x+20)*exp(4))*ln(-ln(x)+x-4)+(-3*x**3+4*x)*exp(4)*ln(x)+(3*x**4-12* x**3-4*x**2+17*x-1)*exp(4))/((ln(x)-x+4)*ln(-ln(x)+x-4)+x*ln(x)-x**2+4*x), x)
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) + 5 \, x e^{4} \]
integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x) +(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^ 3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x^ 4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2 +4*x),x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x^{3} e^{4} - x e^{4} \log \left (x + \log \left (x - \log \left (x\right ) - 4\right )\right ) + 5 \, x e^{4} \]
integrate((((-exp(4)*log(x)+(x-4)*exp(4))*log(-log(x)+x-4)-x*exp(4)*log(x) +(x^2-4*x)*exp(4))*log(log(-log(x)+x-4)+x)+((-3*x^2+5)*exp(4)*log(x)+(3*x^ 3-12*x^2-5*x+20)*exp(4))*log(-log(x)+x-4)+(-3*x^3+4*x)*exp(4)*log(x)+(3*x^ 4-12*x^3-4*x^2+17*x-1)*exp(4))/((log(x)-x+4)*log(-log(x)+x-4)+x*log(x)-x^2 +4*x),x, algorithm=\
Time = 9.61 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {e^4 \left (-1+17 x-4 x^2-12 x^3+3 x^4\right )+e^4 \left (4 x-3 x^3\right ) \log (x)+\left (e^4 \left (20-5 x-12 x^2+3 x^3\right )+e^4 \left (5-3 x^2\right ) \log (x)\right ) \log (-4+x-\log (x))+\left (e^4 \left (-4 x+x^2\right )-e^4 x \log (x)+\left (e^4 (-4+x)-e^4 \log (x)\right ) \log (-4+x-\log (x))\right ) \log (x+\log (-4+x-\log (x)))}{4 x-x^2+x \log (x)+(4-x+\log (x)) \log (-4+x-\log (x))} \, dx=-x\,{\mathrm {e}}^4\,\left (\ln \left (x+\ln \left (x-\ln \left (x\right )-4\right )\right )+x^2-5\right ) \]
int(-(log(x + log(x - log(x) - 4))*(exp(4)*(4*x - x^2) - log(x - log(x) - 4)*(exp(4)*(x - 4) - exp(4)*log(x)) + x*exp(4)*log(x)) + exp(4)*(4*x^2 - 1 7*x + 12*x^3 - 3*x^4 + 1) + log(x - log(x) - 4)*(exp(4)*(5*x + 12*x^2 - 3* x^3 - 20) + exp(4)*log(x)*(3*x^2 - 5)) - exp(4)*log(x)*(4*x - 3*x^3))/(4*x + log(x - log(x) - 4)*(log(x) - x + 4) + x*log(x) - x^2),x)