Integrand size = 109, antiderivative size = 25 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=e^{e^{2 x}+\left (-1+\log \left (4+e^x-\frac {x^2}{6}\right )\right )^2} \]
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {36 e^{1+e^{2 x}+\log ^2\left (4+e^x-\frac {x^2}{6}\right )}}{\left (24+6 e^x-x^2\right )^2} \]
Integrate[(36*E^(1 + Log[(24 + 6*E^x - x^2)/6]^2)*(E^E^(2*x)*(-12*E^x + 4* x + E^(2*x)*(48 + 12*E^x - 2*x^2)) + E^E^(2*x)*(12*E^x - 4*x)*Log[(24 + 6* E^x - x^2)/6]))/(24 + 6*E^x - x^2)^3,x]
Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {27, 27, 2726}
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.
\(\displaystyle \int \frac {36 e^{\log ^2\left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )+1} \left (e^{e^{2 x}} \left (e^{2 x} \left (-2 x^2+12 e^x+48\right )+4 x-12 e^x\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )\right )}{\left (-x^2+6 e^x+24\right )^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 36 \int -\frac {2 e^{\log ^2\left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )+1} \left (e^{e^{2 x}} \left (-2 x+6 e^x-e^{2 x} \left (-x^2+6 e^x+24\right )\right )-2 e^{e^{2 x}} \left (3 e^x-x\right ) \log \left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )\right )}{\left (-x^2+6 e^x+24\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -72 \int \frac {e^{\log ^2\left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )+1} \left (e^{e^{2 x}} \left (-2 x+6 e^x-e^{2 x} \left (-x^2+6 e^x+24\right )\right )-2 e^{e^{2 x}} \left (3 e^x-x\right ) \log \left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )\right )}{\left (-x^2+6 e^x+24\right )^3}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {36 e^{\log ^2\left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )+e^{2 x}+1}}{\left (-x^2+6 e^x+24\right )^2}\) |
Int[(36*E^(1 + Log[(24 + 6*E^x - x^2)/6]^2)*(E^E^(2*x)*(-12*E^x + 4*x + E^ (2*x)*(48 + 12*E^x - 2*x^2)) + E^E^(2*x)*(12*E^x - 4*x)*Log[(24 + 6*E^x - x^2)/6]))/(24 + 6*E^x - x^2)^3,x]
3.19.33.3.1 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[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 18.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {{\mathrm e}^{{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}+1}}{\left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}}\) | \(32\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{2 x}} {\mathrm e}^{\ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}-2 \ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )+1}\) | \(34\) |
int(((12*exp(x)-4*x)*exp(exp(2*x))*ln(exp(x)-1/6*x^2+4)+((12*exp(x)-2*x^2+ 48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(2*x)))*exp(ln(exp(x)-1/6*x^2+4)^2-2*ln (exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=e^{\left (\log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right )^{2} + e^{\left (2 \, x\right )} - 2 \, \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + 1\right )} \]
integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x) -2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4 )^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 1.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {36 e^{\log {\left (- \frac {x^{2}}{6} + e^{x} + 4 \right )}^{2} + 1} e^{e^{2 x}}}{x^{4} - 12 x^{2} e^{x} - 48 x^{2} + 36 e^{2 x} + 288 e^{x} + 576} \]
integrate(((12*exp(x)-4*x)*exp(exp(2*x))*ln(exp(x)-1/6*x**2+4)+((12*exp(x) -2*x**2+48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(2*x)))*exp(ln(exp(x)-1/6*x**2+ 4)**2-2*ln(exp(x)-1/6*x**2+4)+1)/(6*exp(x)-x**2+24),x)
36*exp(log(-x**2/6 + exp(x) + 4)**2 + 1)*exp(exp(2*x))/(x**4 - 12*x**2*exp (x) - 48*x**2 + 36*exp(2*x) + 288*exp(x) + 576)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (20) = 40\).
Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.92 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {9 \cdot 2^{2 \, \log \left (3\right ) + 2} e^{\left (\log \left (3\right )^{2} + \log \left (2\right )^{2} - 2 \, \log \left (3\right ) \log \left (-x^{2} + 6 \, e^{x} + 24\right ) - 2 \, \log \left (2\right ) \log \left (-x^{2} + 6 \, e^{x} + 24\right ) + \log \left (-x^{2} + 6 \, e^{x} + 24\right )^{2} + e^{\left (2 \, x\right )} + 1\right )}}{x^{4} - 48 \, x^{2} - 12 \, {\left (x^{2} - 24\right )} e^{x} + 36 \, e^{\left (2 \, x\right )} + 576} \]
integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x) -2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4 )^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm=\
9*2^(2*log(3) + 2)*e^(log(3)^2 + log(2)^2 - 2*log(3)*log(-x^2 + 6*e^x + 24 ) - 2*log(2)*log(-x^2 + 6*e^x + 24) + log(-x^2 + 6*e^x + 24)^2 + e^(2*x) + 1)/(x^4 - 48*x^2 - 12*(x^2 - 24)*e^x + 36*e^(2*x) + 576)
\[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\int { \frac {2 \, {\left (2 \, {\left (x - 3 \, e^{x}\right )} e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + {\left ({\left (x^{2} - 6 \, e^{x} - 24\right )} e^{\left (2 \, x\right )} - 2 \, x + 6 \, e^{x}\right )} e^{\left (e^{\left (2 \, x\right )}\right )}\right )} e^{\left (\log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right )^{2} - 2 \, \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + 1\right )}}{x^{2} - 6 \, e^{x} - 24} \,d x } \]
integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x) -2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4 )^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm=\
integrate(2*(2*(x - 3*e^x)*e^(e^(2*x))*log(-1/6*x^2 + e^x + 4) + ((x^2 - 6 *e^x - 24)*e^(2*x) - 2*x + 6*e^x)*e^(e^(2*x)))*e^(log(-1/6*x^2 + e^x + 4)^ 2 - 2*log(-1/6*x^2 + e^x + 4) + 1)/(x^2 - 6*e^x - 24), x)
Time = 14.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {\mathrm {e}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{{\ln \left ({\mathrm {e}}^x-\frac {x^2}{6}+4\right )}^2}}{{\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x-\frac {x^2\,{\mathrm {e}}^x}{3}-\frac {4\,x^2}{3}+\frac {x^4}{36}+16} \]
int((exp(log(exp(x) - x^2/6 + 4)^2 - 2*log(exp(x) - x^2/6 + 4) + 1)*(exp(e xp(2*x))*(4*x - 12*exp(x) + exp(2*x)*(12*exp(x) - 2*x^2 + 48)) - exp(exp(2 *x))*log(exp(x) - x^2/6 + 4)*(4*x - 12*exp(x))))/(6*exp(x) - x^2 + 24),x)