Integrand size = 71, antiderivative size = 29 \[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=4-2^{-2 x} e^{e^{-4+\frac {1}{3} \left (\frac {2}{x}-2 x\right )}} x \] Output:
4-exp(exp(2/3/x-2/3*x-4))/exp(x*ln(2))^2*x
Time = 0.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=-\frac {2^{-1-2 x} e^{e^{-4+\frac {2}{3 x}-\frac {2 x}{3}}} x \log (4)}{\log (2)} \] Input:
Integrate[(E^E^((2 - 12*x - 2*x^2)/(3*x))*(-3*x + E^((2 - 12*x - 2*x^2)/(3 *x))*(2 + 2*x^2) + 6*x^2*Log[2]))/(3*2^(2*x)*x),x]
Output:
-((2^(-1 - 2*x)*E^E^(-4 + 2/(3*x) - (2*x)/3)*x*Log[4])/Log[2])
Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {2^{-2 x} e^{e^{\frac {-2 x^2-12 x+2}{3 x}}} \left (e^{\frac {-2 x^2-12 x+2}{3 x}} \left (2 x^2+2\right )+6 x^2 \log (2)-3 x\right )}{3 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {2^{-2 x} e^{e^{\frac {2 \left (-x^2-6 x+1\right )}{3 x}}} \left (-6 \log (2) x^2+3 x-2 e^{\frac {2 \left (-x^2-6 x+1\right )}{3 x}} \left (x^2+1\right )\right )}{x}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {2^{-2 x} e^{e^{\frac {2 \left (-x^2-6 x+1\right )}{3 x}}} \left (-6 \log (2) x^2+3 x-2 e^{\frac {2 \left (-x^2-6 x+1\right )}{3 x}} \left (x^2+1\right )\right )}{x}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -2^{-2 x} e^{e^{\frac {2 \left (-x^2-6 x+1\right )}{3 x}}} x\) |
Input:
Int[(E^E^((2 - 12*x - 2*x^2)/(3*x))*(-3*x + E^((2 - 12*x - 2*x^2)/(3*x))*( 2 + 2*x^2) + 6*x^2*Log[2]))/(3*2^(2*x)*x),x]
Output:
-((E^E^((2*(1 - 6*x - x^2))/(3*x))*x)/2^(2*x))
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 = 1.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-x 2^{-2 x} {\mathrm e}^{{\mathrm e}^{-\frac {2 \left (x^{2}+6 x -1\right )}{3 x}}}\) | \(24\) |
parallelrisch | \(-{\mathrm e}^{{\mathrm e}^{-\frac {2 \left (x^{2}+6 x -1\right )}{3 x}}} x \,{\mathrm e}^{-2 x \ln \left (2\right )}\) | \(26\) |
Input:
int(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*ln(2)-3*x)*exp(exp(1/3 *(-2*x^2-12*x+2)/x))/x/exp(x*ln(2))^2,x,method=_RETURNVERBOSE)
Output:
-x/(2^x)^2*exp(exp(-2/3*(x^2+6*x-1)/x))
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=-\frac {x e^{\left (e^{\left (-\frac {2 \, {\left (x^{2} + 6 \, x - 1\right )}}{3 \, x}\right )}\right )}}{2^{2 \, x}} \] Input:
integrate(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*log(2)-3*x)*exp( exp(1/3*(-2*x^2-12*x+2)/x))/x/exp(x*log(2))^2,x, algorithm="fricas")
Output:
-x*e^(e^(-2/3*(x^2 + 6*x - 1)/x))/2^(2*x)
Time = 70.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=- x e^{- 2 x \log {\left (2 \right )}} e^{e^{\frac {- \frac {2 x^{2}}{3} - 4 x + \frac {2}{3}}{x}}} \] Input:
integrate(1/3*((2*x**2+2)*exp(1/3*(-2*x**2-12*x+2)/x)+6*x**2*ln(2)-3*x)*ex p(exp(1/3*(-2*x**2-12*x+2)/x))/x/exp(x*ln(2))**2,x)
Output:
-x*exp(-2*x*log(2))*exp(exp((-2*x**2/3 - 4*x + 2/3)/x))
\[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=\int { \frac {{\left (6 \, x^{2} \log \left (2\right ) + 2 \, {\left (x^{2} + 1\right )} e^{\left (-\frac {2 \, {\left (x^{2} + 6 \, x - 1\right )}}{3 \, x}\right )} - 3 \, x\right )} e^{\left (e^{\left (-\frac {2 \, {\left (x^{2} + 6 \, x - 1\right )}}{3 \, x}\right )}\right )}}{3 \cdot 2^{2 \, x} x} \,d x } \] Input:
integrate(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*log(2)-3*x)*exp( exp(1/3*(-2*x^2-12*x+2)/x))/x/exp(x*log(2))^2,x, algorithm="maxima")
Output:
1/3*integrate((6*x^2*log(2) + 2*(x^2 + 1)*e^(-2/3*(x^2 + 6*x - 1)/x) - 3*x )*e^(e^(-2/3*(x^2 + 6*x - 1)/x))/(2^(2*x)*x), x)
\[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=\int { \frac {{\left (6 \, x^{2} \log \left (2\right ) + 2 \, {\left (x^{2} + 1\right )} e^{\left (-\frac {2 \, {\left (x^{2} + 6 \, x - 1\right )}}{3 \, x}\right )} - 3 \, x\right )} e^{\left (e^{\left (-\frac {2 \, {\left (x^{2} + 6 \, x - 1\right )}}{3 \, x}\right )}\right )}}{3 \cdot 2^{2 \, x} x} \,d x } \] Input:
integrate(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*log(2)-3*x)*exp( exp(1/3*(-2*x^2-12*x+2)/x))/x/exp(x*log(2))^2,x, algorithm="giac")
Output:
integrate(1/3*(6*x^2*log(2) + 2*(x^2 + 1)*e^(-2/3*(x^2 + 6*x - 1)/x) - 3*x )*e^(e^(-2/3*(x^2 + 6*x - 1)/x))/(2^(2*x)*x), x)
Time = 4.56 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=-{\left (\frac {1}{4}\right )}^x\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {2\,x}{3}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{\frac {2}{3\,x}}} \] Input:
int((exp(exp(-(4*x + (2*x^2)/3 - 2/3)/x))*exp(-2*x*log(2))*(6*x^2*log(2) - 3*x + exp(-(4*x + (2*x^2)/3 - 2/3)/x)*(2*x^2 + 2)))/(3*x),x)
Output:
-(1/4)^x*x*exp(exp(-(2*x)/3)*exp(-4)*exp(2/(3*x)))
\[ \int \frac {2^{-2 x} e^{e^{\frac {2-12 x-2 x^2}{3 x}}} \left (-3 x+e^{\frac {2-12 x-2 x^2}{3 x}} \left (2+2 x^2\right )+6 x^2 \log (2)\right )}{3 x} \, dx=\int \frac {\left (\left (2 x^{2}+2\right ) {\mathrm e}^{\frac {-2 x^{2}-12 x +2}{3 x}}+6 \,\mathrm {log}\left (2\right ) x^{2}-3 x \right ) {\mathrm e}^{{\mathrm e}^{\frac {-2 x^{2}-12 x +2}{3 x}}}}{3 x \left ({\mathrm e}^{\mathrm {log}\left (2\right ) x}\right )^{2}}d x \] Input:
int(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*log(2)-3*x)*exp(exp(1/ 3*(-2*x^2-12*x+2)/x))/x/exp(x*log(2))^2,x)
Output:
int(1/3*((2*x^2+2)*exp(1/3*(-2*x^2-12*x+2)/x)+6*x^2*log(2)-3*x)*exp(exp(1/ 3*(-2*x^2-12*x+2)/x))/x/exp(x*log(2))^2,x)