Integrand size = 89, antiderivative size = 27 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{-x+\frac {x \left (5+x+(4-2 x-\log (x))^2\right )}{-3+x}} \]
Time = 5.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\frac {x \left (4 \left (6-4 x+x^2\right )+\log ^2(x)\right )}{-3+x}} x^{\frac {4 (-2+x) x}{-3+x}} \]
Integrate[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[ x]^2)/(-3 + x))*(-48 + 76*x - 48*x^2 + 8*x^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Log[x]^2))/(9 - 6*x + 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.
\(\displaystyle \int \frac {\left (8 x^3-48 x^2+\left (4 x^2-22 x+18\right ) \log (x)+76 x-3 \log ^2(x)-48\right ) \exp \left (\frac {4 x^3-15 x^2+\left (4 x^2-8 x\right ) \log (x)+21 x+x \log ^2(x)}{x-3}-x\right )}{x^2-6 x+9} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int -\frac {\exp \left (-x-\frac {4 x^3-15 x^2+\log ^2(x) x+21 x-4 \left (2 x-x^2\right ) \log (x)}{3-x}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{4 (3-x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {\exp \left (-x-\frac {4 x^3-15 x^2+\log ^2(x) x+21 x-4 \left (2 x-x^2\right ) \log (x)}{3-x}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{(3-x)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \left (-8 x^3+48 x^2-76 x+3 \log ^2(x)-2 \left (2 x^2-11 x+9\right ) \log (x)+48\right )}{(3-x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (-\frac {8 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x^3}{(x-3)^2}+\frac {48 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x^2}{(x-3)^2}-\frac {76 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) x}{(x-3)^2}+\frac {3 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log ^2(x)}{(x-3)^2}-\frac {2 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) (x-1) (2 x-9) \log (x)}{(x-3)^2}+\frac {48 \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{(x-3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -36 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{(x-3)^2}dx+4 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right )}{x-3}dx+8 \int \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) xdx+4 \int \exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)dx-12 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)}{(x-3)^2}dx+2 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log (x)}{x-3}dx-3 \int \frac {\exp \left (\frac {x \left (4 x^2+4 \log (x) x-16 x+\log ^2(x)-8 \log (x)+24\right )}{x-3}\right ) \log ^2(x)}{(x-3)^2}dx\) |
Int[(E^(-x + (21*x - 15*x^2 + 4*x^3 + (-8*x + 4*x^2)*Log[x] + x*Log[x]^2)/ (-3 + x))*(-48 + 76*x - 48*x^2 + 8*x^3 + (18 - 22*x + 4*x^2)*Log[x] - 3*Lo g[x]^2))/(9 - 6*x + x^2),x]
3.12.89.3.1 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_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 1.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \({\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-16 x +24\right )}{-3+x}}\) | \(32\) |
parallelrisch | \({\mathrm e}^{-x} {\mathrm e}^{\frac {x \left (\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}-8 \ln \left (x \right )-15 x +21\right )}{-3+x}}\) | \(37\) |
int((-3*ln(x)^2+(4*x^2-22*x+18)*ln(x)+8*x^3-48*x^2+76*x-48)*exp((x*ln(x)^2 +(4*x^2-8*x)*ln(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x),x,method= _RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3} + x \log \left (x\right )^{2} - 16 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 24 \, x}{x - 3}\right )} \]
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm=\
Time = 35.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{- x} e^{\frac {4 x^{3} - 15 x^{2} + x \log {\left (x \right )}^{2} + 21 x + \left (4 x^{2} - 8 x\right ) \log {\left (x \right )}}{x - 3}} \]
integrate((-3*ln(x)**2+(4*x**2-22*x+18)*ln(x)+8*x**3-48*x**2+76*x-48)*exp( (x*ln(x)**2+(4*x**2-8*x)*ln(x)+4*x**3-15*x**2+21*x)/(-3+x))/(x**2-6*x+9)/e xp(x),x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=x^{4} e^{\left (4 \, x^{2} + 4 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x + \frac {3 \, \log \left (x\right )^{2}}{x - 3} + \frac {12 \, \log \left (x\right )}{x - 3} + \frac {36}{x - 3} + 12\right )} \]
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm=\
x^4*e^(4*x^2 + 4*x*log(x) + log(x)^2 - 4*x + 3*log(x)^2/(x - 3) + 12*log(x )/(x - 3) + 36/(x - 3) + 12)
Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=e^{\left (\frac {4 \, x^{3} + 4 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 16 \, x^{2} - 8 \, x \log \left (x\right ) + 24 \, x}{x - 3}\right )} \]
integrate((-3*log(x)^2+(4*x^2-22*x+18)*log(x)+8*x^3-48*x^2+76*x-48)*exp((x *log(x)^2+(4*x^2-8*x)*log(x)+4*x^3-15*x^2+21*x)/(-3+x))/(x^2-6*x+9)/exp(x) ,x, algorithm=\
Time = 13.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {e^{-x+\frac {21 x-15 x^2+4 x^3+\left (-8 x+4 x^2\right ) \log (x)+x \log ^2(x)}{-3+x}} \left (-48+76 x-48 x^2+8 x^3+\left (18-22 x+4 x^2\right ) \log (x)-3 \log ^2(x)\right )}{9-6 x+x^2} \, dx=\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {21\,x}{x-3}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x\right )}^2}{x-3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x-3}}\,{\mathrm {e}}^{-\frac {15\,x^2}{x-3}}}{x^{\frac {4\,\left (2\,x-x^2\right )}{x-3}}} \]
int((exp(-x)*exp((21*x + x*log(x)^2 - log(x)*(8*x - 4*x^2) - 15*x^2 + 4*x^ 3)/(x - 3))*(76*x - 3*log(x)^2 + log(x)*(4*x^2 - 22*x + 18) - 48*x^2 + 8*x ^3 - 48))/(x^2 - 6*x + 9),x)