Integrand size = 86, antiderivative size = 32 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 ((-1+x) x+\log (x))}{5-\frac {3-x}{20-x}}} \]
Time = 5.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {e^2 x \left (20-21 x+x^2\right )}{-97+4 x}} x^{\frac {e^2 (-20+x)}{-97+4 x}} \]
Integrate[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4* x))*(E^2*(1940 - 2117*x + 4078*x^2 - 375*x^3 + 8*x^4) - 17*E^2*x*Log[x]))/ (9409*x - 776*x^2 + 16*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.
\(\displaystyle \int \frac {\left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right ) \exp \left (\frac {e^2 \left (x^3-21 x^2+20 x\right )+e^2 (x-20) \log (x)}{4 x-97}\right )}{16 x^3-776 x^2+9409 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right ) \exp \left (\frac {e^2 \left (x^3-21 x^2+20 x\right )+e^2 (x-20) \log (x)}{4 x-97}\right )}{x \left (16 x^2-776 x+9409\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 64 \int \frac {e^{-\frac {e^2 \left (x^3-21 x^2+20 x\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right )}{64 (97-4 x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {e^{-\frac {e^2 \left (x^3-21 x^2+20 x\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (e^2 \left (8 x^4-375 x^3+4078 x^2-2117 x+1940\right )-17 e^2 x \log (x)\right )}{(97-4 x)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1} \left (8 x^4-375 x^3+4078 x^2-2117 x-17 x \log (x)+1940\right )}{(97-4 x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1940 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1}}{(4 x-97)^2}+\frac {4078 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+1}}{(4 x-97)^2}-\frac {375 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+2}}{(4 x-97)^2}+\frac {8 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+3}}{(4 x-97)^2}-\frac {2117 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}-\frac {17 e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}} \log (x)}{(4 x-97)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 1940 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}-1}}{(4 x-97)^2}dx+4078 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+1}}{(4 x-97)^2}dx-375 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+2}}{(4 x-97)^2}dx+8 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}+3}}{(4 x-97)^2}dx-2117 \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}dx+17 \int \frac {\int \frac {e^{\frac {e^2 x \left (x^2-21 x+20\right )}{4 x-97}+2} x^{\frac {e^2 (x-20)}{4 x-97}}}{(97-4 x)^2}dx}{x}dx-17 \log (x) \int \frac {e^{2-\frac {e^2 x \left (x^2-21 x+20\right )}{97-4 x}} x^{\frac {e^2 (20-x)}{97-4 x}}}{(4 x-97)^2}dx\) |
Int[(E^((E^2*(20*x - 21*x^2 + x^3) + E^2*(-20 + x)*Log[x])/(-97 + 4*x))*(E ^2*(1940 - 2117*x + 4078*x^2 - 375*x^3 + 8*x^4) - 17*E^2*x*Log[x]))/(9409* x - 776*x^2 + 16*x^3),x]
3.8.63.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, 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 = 1.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x -20\right ) \left (x^{2}+\ln \left (x \right )-x \right )}{4 x -97}}\) | \(24\) |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{2} \left (x^{3}+x \ln \left (x \right )-21 x^{2}-20 \ln \left (x \right )+20 x \right )}{4 x -97}}\) | \(32\) |
norman | \(\frac {4 x \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}-97 \,{\mathrm e}^{\frac {\left (x -20\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{3}-21 x^{2}+20 x \right ) {\mathrm e}^{2}}{4 x -97}}}{4 x -97}\) | \(81\) |
int((-17*x*exp(2)*ln(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2))*exp(( (x-20)*exp(2)*ln(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-776*x^2+94 09*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {{\left (x - 20\right )} e^{2} \log \left (x\right ) + {\left (x^{3} - 21 \, x^{2} + 20 \, x\right )} e^{2}}{4 \, x - 97}\right )} \]
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\frac {\left (x - 20\right ) e^{2} \log {\left (x \right )} + \left (x^{3} - 21 x^{2} + 20 x\right ) e^{2}}{4 x - 97}} \]
integrate((-17*x*exp(2)*ln(x)+(8*x**4-375*x**3+4078*x**2-2117*x+1940)*exp( 2))*exp(((x-20)*exp(2)*ln(x)+(x**3-21*x**2+20*x)*exp(2))/(4*x-97))/(16*x** 3-776*x**2+9409*x),x)
Time = 19.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {1}{4} \, x^{2} e^{2} + \frac {13}{16} \, x e^{2} + \frac {1}{4} \, e^{2} \log \left (x\right ) + \frac {17 \, e^{2} \log \left (x\right )}{4 \, {\left (4 \, x - 97\right )}} + \frac {153357 \, e^{2}}{64 \, {\left (4 \, x - 97\right )}} + \frac {1581}{64} \, e^{2}\right )} \]
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm=\
e^(1/4*x^2*e^2 + 13/16*x*e^2 + 1/4*e^2*log(x) + 17/4*e^2*log(x)/(4*x - 97) + 153357/64*e^2/(4*x - 97) + 1581/64*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=e^{\left (\frac {x^{3} e^{2}}{4 \, x - 97} - \frac {21 \, x^{2} e^{2}}{4 \, x - 97} + \frac {x e^{2} \log \left (x\right )}{4 \, x - 97} + \frac {20 \, x e^{2}}{4 \, x - 97} - \frac {20 \, e^{2} \log \left (x\right )}{4 \, x - 97}\right )} \]
integrate((-17*x*exp(2)*log(x)+(8*x^4-375*x^3+4078*x^2-2117*x+1940)*exp(2) )*exp(((x-20)*exp(2)*log(x)+(x^3-21*x^2+20*x)*exp(2))/(4*x-97))/(16*x^3-77 6*x^2+9409*x),x, algorithm=\
e^(x^3*e^2/(4*x - 97) - 21*x^2*e^2/(4*x - 97) + x*e^2*log(x)/(4*x - 97) + 20*x*e^2/(4*x - 97) - 20*e^2*log(x)/(4*x - 97))
Time = 12.88 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {e^2 \left (20 x-21 x^2+x^3\right )+e^2 (-20+x) \log (x)}{-97+4 x}} \left (e^2 \left (1940-2117 x+4078 x^2-375 x^3+8 x^4\right )-17 e^2 x \log (x)\right )}{9409 x-776 x^2+16 x^3} \, dx=\frac {{\mathrm {e}}^{\frac {20\,x\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{\frac {x^3\,{\mathrm {e}}^2}{4\,x-97}}\,{\mathrm {e}}^{-\frac {21\,x^2\,{\mathrm {e}}^2}{4\,x-97}}}{x^{\frac {20\,{\mathrm {e}}^2-x\,{\mathrm {e}}^2}{4\,x-97}}} \]
int((exp((exp(2)*(20*x - 21*x^2 + x^3) + exp(2)*log(x)*(x - 20))/(4*x - 97 ))*(exp(2)*(4078*x^2 - 2117*x - 375*x^3 + 8*x^4 + 1940) - 17*x*exp(2)*log( x)))/(9409*x - 776*x^2 + 16*x^3),x)