Integrand size = 165, antiderivative size = 22 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {3 x \left (-1+\frac {e x}{-x+\log (-4+x)}\right )}{\log (x)} \]
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 x (x+e x-\log (-4+x))}{(x-\log (-4+x)) \log (x)} \]
Integrate[(-12*x^2 + 3*x^3 + E*(-12*x^2 + 3*x^3) + (24*x - 6*x^2 + E*(12*x - 3*x^2))*Log[-4 + x] + (-12 + 3*x)*Log[-4 + x]^2 + (12*x^2 - 3*x^3 + E*( 9*x^2 - 3*x^3) + (-24*x + 6*x^2 + E*(-24*x + 6*x^2))*Log[-4 + x] + (12 - 3 *x)*Log[-4 + x]^2)*Log[x])/((-4*x^2 + x^3 + (8*x - 2*x^2)*Log[-4 + x] + (- 4 + x)*Log[-4 + x]^2)*Log[x]^2),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.
| \(\displaystyle \int \frac {3 x^3-12 x^2+\left (-6 x^2+e \left (12 x-3 x^2\right )+24 x\right ) \log (x-4)+e \left (3 x^3-12 x^2\right )+\left (-3 x^3+12 x^2+\left (6 x^2+e \left (6 x^2-24 x\right )-24 x\right ) \log (x-4)+e \left (9 x^2-3 x^3\right )+(12-3 x) \log ^2(x-4)\right ) \log (x)+(3 x-12) \log ^2(x-4)}{\left (x^3-4 x^2+\left (8 x-2 x^2\right ) \log (x-4)+(x-4) \log ^2(x-4)\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 \left (x^2 ((e (x-3)+x-4) \log (x)-(1+e) (x-4))+(x-4) \log ^2(x-4) (\log (x)-1)-(x-4) x \log (x-4) (2 (1+e) \log (x)-e-2)\right )}{(4-x) (x-\log (x-4))^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {((1+e) (4-x)-(e (3-x)-x+4) \log (x)) x^2-(4-x) \log (x-4) (-2 (1+e) \log (x)+e+2) x+(4-x) \log ^2(x-4) (1-\log (x))}{(4-x) (x-\log (x-4))^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (\frac {(1+e) x-\log (x-4)}{(x-\log (x-4)) \log ^2(x)}+\frac {(1+e) x^3-2 (1+e) \log (x-4) x^2-4 \left (1+\frac {3 e}{4}\right ) x^2+\log ^2(x-4) x+8 (1+e) \log (x-4) x-4 \log ^2(x-4)}{(4-x) (x-\log (x-4))^2 \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (-(1+e) \int \frac {x^2}{(x-\log (x-4))^2 \log (x)}dx+(1+e) \int \frac {x}{(x-\log (x-4)) \log ^2(x)}dx-\int \frac {\log (x-4)}{(x-\log (x-4)) \log ^2(x)}dx-\int \frac {\log ^2(x-4)}{(x-\log (x-4))^2 \log (x)}dx+4 (4+3 e) \int \frac {1}{(x-\log (x-4))^2 \log (x)}dx-16 (1+e) \int \frac {1}{(x-\log (x-4))^2 \log (x)}dx+16 (4+3 e) \int \frac {1}{(x-4) (x-\log (x-4))^2 \log (x)}dx-64 (1+e) \int \frac {1}{(x-4) (x-\log (x-4))^2 \log (x)}dx+(4+3 e) \int \frac {x}{(x-\log (x-4))^2 \log (x)}dx-4 (1+e) \int \frac {x}{(x-\log (x-4))^2 \log (x)}dx+2 (1+e) \int \frac {x \log (x-4)}{(x-\log (x-4))^2 \log (x)}dx\right )\) |
Int[(-12*x^2 + 3*x^3 + E*(-12*x^2 + 3*x^3) + (24*x - 6*x^2 + E*(12*x - 3*x ^2))*Log[-4 + x] + (-12 + 3*x)*Log[-4 + x]^2 + (12*x^2 - 3*x^3 + E*(9*x^2 - 3*x^3) + (-24*x + 6*x^2 + E*(-24*x + 6*x^2))*Log[-4 + x] + (12 - 3*x)*Lo g[-4 + x]^2)*Log[x])/((-4*x^2 + x^3 + (8*x - 2*x^2)*Log[-4 + x] + (-4 + x) *Log[-4 + x]^2)*Log[x]^2),x]
3.27.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
| risch | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
| parts | \(-\frac {3 x}{\ln \left (x \right )}-\frac {3 \,{\mathrm e} x^{2}}{\ln \left (x \right ) \left (-\ln \left (x -4\right )+x \right )}\) | \(30\) |
| parallelrisch | \(-\frac {3 x^{2} {\mathrm e}+3 x^{2}-3 x \ln \left (x -4\right )}{\left (-\ln \left (x -4\right )+x \right ) \ln \left (x \right )}\) | \(37\) |
int((((-3*x+12)*ln(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*ln(x-4)+(-3*x^3 +9*x^2)*exp(1)-3*x^3+12*x^2)*ln(x)+(3*x-12)*ln(x-4)^2+((-3*x^2+12*x)*exp(1 )-6*x^2+24*x)*ln(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x-4)*ln(x-4)^2 +(-2*x^2+8*x)*ln(x-4)+x^3-4*x^2)/ln(x)^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{{\left (x - \log \left (x - 4\right )\right )} \log \left (x\right )} \]
integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4) +(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+1 2*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x-4 )*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {3 e x^{2}}{- x \log {\left (x \right )} + \log {\left (x \right )} \log {\left (x - 4 \right )}} - \frac {3 x}{\log {\left (x \right )}} \]
integrate((((-3*x+12)*ln(x-4)**2+((6*x**2-24*x)*exp(1)+6*x**2-24*x)*ln(x-4 )+(-3*x**3+9*x**2)*exp(1)-3*x**3+12*x**2)*ln(x)+(3*x-12)*ln(x-4)**2+((-3*x **2+12*x)*exp(1)-6*x**2+24*x)*ln(x-4)+(3*x**3-12*x**2)*exp(1)+3*x**3-12*x* *2)/((x-4)*ln(x-4)**2+(-2*x**2+8*x)*ln(x-4)+x**3-4*x**2)/ln(x)**2,x)
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} {\left (e + 1\right )} - x \log \left (x - 4\right )\right )}}{x \log \left (x\right ) - \log \left (x - 4\right ) \log \left (x\right )} \]
integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4) +(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+1 2*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x-4 )*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x^{2} e + x^{2} - x \log \left (x - 4\right )\right )}}{x \log \left (x\right ) - \log \left (x - 4\right ) \log \left (x\right )} \]
integrate((((-3*x+12)*log(x-4)^2+((6*x^2-24*x)*exp(1)+6*x^2-24*x)*log(x-4) +(-3*x^3+9*x^2)*exp(1)-3*x^3+12*x^2)*log(x)+(3*x-12)*log(x-4)^2+((-3*x^2+1 2*x)*exp(1)-6*x^2+24*x)*log(x-4)+(3*x^3-12*x^2)*exp(1)+3*x^3-12*x^2)/((x-4 )*log(x-4)^2+(-2*x^2+8*x)*log(x-4)+x^3-4*x^2)/log(x)^2,x, algorithm=\
Time = 10.22 (sec) , antiderivative size = 402, normalized size of antiderivative = 18.27 \[ \int \frac {-12 x^2+3 x^3+e \left (-12 x^2+3 x^3\right )+\left (24 x-6 x^2+e \left (12 x-3 x^2\right )\right ) \log (-4+x)+(-12+3 x) \log ^2(-4+x)+\left (12 x^2-3 x^3+e \left (9 x^2-3 x^3\right )+\left (-24 x+6 x^2+e \left (-24 x+6 x^2\right )\right ) \log (-4+x)+(12-3 x) \log ^2(-4+x)\right ) \log (x)}{\left (-4 x^2+x^3+\left (8 x-2 x^2\right ) \log (-4+x)+(-4+x) \log ^2(-4+x)\right ) \log ^2(x)} \, dx=\frac {\frac {3\,x\,\ln \left (x\right )\,\left (75\,x-100\,\mathrm {e}+60\,x\,\mathrm {e}-15\,x^2\,\mathrm {e}+x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}-\frac {3\,x\,\left (20\,\mathrm {e}-10\,x-8\,x\,\mathrm {e}+x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \left (x\right )}^2\,\left (75\,x-200\,\mathrm {e}+160\,x\,\mathrm {e}-30\,x^2\,\mathrm {e}+2\,x^3\,\mathrm {e}-15\,x^2+x^3-125\right )}{2\,{\left (x-5\right )}^3}}{\ln \left (x\right )}+\frac {\frac {3\,x^2\,\left (4\,\mathrm {e}-x\,\mathrm {e}-3\,\mathrm {e}\,\ln \left (x\right )+x\,\mathrm {e}\,\ln \left (x\right )\right )}{{\ln \left (x\right )}^2\,\left (x-5\right )}+\frac {3\,\ln \left (x-4\right )\,\left (x\,\mathrm {e}-2\,x\,\mathrm {e}\,\ln \left (x\right )\right )\,\left (x-4\right )}{{\ln \left (x\right )}^2\,\left (x-5\right )}}{x-\ln \left (x-4\right )}+\ln \left (x\right )\,\left (45\,\mathrm {e}+\frac {45}{2}\right )-\frac {225\,x\,\mathrm {e}-75\,x^2\,\mathrm {e}}{2\,x^3-30\,x^2+150\,x-250}+\frac {\frac {3\,x\,\mathrm {e}\,\left (x-4\right )}{x-5}-\frac {3\,x\,\ln \left (x\right )\,\left (60\,\mathrm {e}-10\,x-28\,x\,\mathrm {e}+3\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}+\frac {3\,x\,{\ln \left (x\right )}^2\,\left (40\,\mathrm {e}-10\,x-20\,x\,\mathrm {e}+2\,x^2\,\mathrm {e}+x^2+25\right )}{2\,{\left (x-5\right )}^2}}{{\ln \left (x\right )}^2}-x\,\left (\frac {9\,\mathrm {e}}{2}+3\right )+\frac {\ln \left (x\right )\,\left (\left (-3\,\mathrm {e}-\frac {3}{2}\right )\,x^4+\left (435\,\mathrm {e}+225\right )\,x^2+\left (-3075\,\mathrm {e}-1500\right )\,x+5625\,\mathrm {e}+\frac {5625}{2}\right )}{x^3-15\,x^2+75\,x-125} \]
int(-(log(x)*(log(x - 4)^2*(3*x - 12) + log(x - 4)*(24*x + exp(1)*(24*x - 6*x^2) - 6*x^2) - exp(1)*(9*x^2 - 3*x^3) - 12*x^2 + 3*x^3) - log(x - 4)^2* (3*x - 12) - log(x - 4)*(24*x + exp(1)*(12*x - 3*x^2) - 6*x^2) + exp(1)*(1 2*x^2 - 3*x^3) + 12*x^2 - 3*x^3)/(log(x)^2*(log(x - 4)*(8*x - 2*x^2) - 4*x ^2 + x^3 + log(x - 4)^2*(x - 4))),x)
((3*x*log(x)*(75*x - 100*exp(1) + 60*x*exp(1) - 15*x^2*exp(1) + x^3*exp(1) - 15*x^2 + x^3 - 125))/(2*(x - 5)^3) - (3*x*(20*exp(1) - 10*x - 8*x*exp(1 ) + x^2*exp(1) + x^2 + 25))/(2*(x - 5)^2) + (3*x*log(x)^2*(75*x - 200*exp( 1) + 160*x*exp(1) - 30*x^2*exp(1) + 2*x^3*exp(1) - 15*x^2 + x^3 - 125))/(2 *(x - 5)^3))/log(x) + ((3*x^2*(4*exp(1) - x*exp(1) - 3*exp(1)*log(x) + x*e xp(1)*log(x)))/(log(x)^2*(x - 5)) + (3*log(x - 4)*(x*exp(1) - 2*x*exp(1)*l og(x))*(x - 4))/(log(x)^2*(x - 5)))/(x - log(x - 4)) + log(x)*(45*exp(1) + 45/2) - (225*x*exp(1) - 75*x^2*exp(1))/(150*x - 30*x^2 + 2*x^3 - 250) + ( (3*x*exp(1)*(x - 4))/(x - 5) - (3*x*log(x)*(60*exp(1) - 10*x - 28*x*exp(1) + 3*x^2*exp(1) + x^2 + 25))/(2*(x - 5)^2) + (3*x*log(x)^2*(40*exp(1) - 10 *x - 20*x*exp(1) + 2*x^2*exp(1) + x^2 + 25))/(2*(x - 5)^2))/log(x)^2 - x*( (9*exp(1))/2 + 3) + (log(x)*(5625*exp(1) - x^4*(3*exp(1) + 3/2) + x^2*(435 *exp(1) + 225) - x*(3075*exp(1) + 1500) + 5625/2))/(75*x - 15*x^2 + x^3 - 125)