Integrand size = 42, antiderivative size = 31 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=e^{-\frac {x^2}{3}} \left (3 e^{3 \left (2 x+\frac {24}{\log (\log (4))}\right )}-x\right ) \] Output:
(3*exp(6*x+72/ln(2*ln(2)))-x)/exp(1/3*x^2)
Time = 1.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=\frac {1}{3} e^{-\frac {x^2}{3}} \left (9 e^{6 x+\frac {72}{\log (\log (4))}}-3 x\right ) \] Input:
Integrate[(-3 + E^((72 + 6*x*Log[Log[4]])/Log[Log[4]])*(54 - 6*x) + 2*x^2) /(3*E^(x^2/3)),x]
Output:
(9*E^(6*x + 72/Log[Log[4]]) - 3*x)/(3*E^(x^2/3))
Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {27, 25, 7293, 2009}
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.
\(\displaystyle \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (2 x^2+(54-6 x) e^{\frac {6 x \log (\log (4))+72}{\log (\log (4))}}-3\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -e^{-\frac {x^2}{3}} \left (-2 x^2-6 e^{\frac {72}{\log (\log (4))}} (9-x) \log ^{\frac {6 x}{\log (\log (4))}}(4)+3\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int e^{-\frac {x^2}{3}} \left (-2 x^2-6 e^{\frac {72}{\log (\log (4))}} (9-x) \log ^{\frac {6 x}{\log (\log (4))}}(4)+3\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (-2 e^{-\frac {x^2}{3}} x^2+3 e^{-\frac {x^2}{3}}+6 e^{-\frac {x^2}{3}+6 x+\frac {72}{\log (\log (4))}} (x-9)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (9 e^{-\frac {x^2}{3}+6 x+\frac {72}{\log (\log (4))}}-3 e^{-\frac {x^2}{3}} x\right )\) |
Input:
Int[(-3 + E^((72 + 6*x*Log[Log[4]])/Log[Log[4]])*(54 - 6*x) + 2*x^2)/(3*E^ (x^2/3)),x]
Output:
(9*E^(6*x - x^2/3 + 72/Log[Log[4]]) - (3*x)/E^(x^2/3))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
default | \(-x \,{\mathrm e}^{-\frac {x^{2}}{3}}+3 \,{\mathrm e}^{-\frac {x^{2}}{3}+6 x +\frac {72}{\ln \left (2 \ln \left (2\right )\right )}}\) | \(32\) |
parts | \(-x \,{\mathrm e}^{-\frac {x^{2}}{3}}+3 \,{\mathrm e}^{-\frac {x^{2}}{3}+6 x +\frac {72}{\ln \left (2 \ln \left (2\right )\right )}}\) | \(32\) |
norman | \(\left (-x +3 \,{\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}\right ) {\mathrm e}^{-\frac {x^{2}}{3}}\) | \(35\) |
parallelrisch | \(\frac {\left (-9 x +27 \,{\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}\right ) {\mathrm e}^{-\frac {x^{2}}{3}}}{9}\) | \(36\) |
risch | \(-x \,{\mathrm e}^{-\frac {x^{2}}{3}}+3 \,{\mathrm e}^{-\frac {x^{2} \ln \left (2\right )+x^{2} \ln \left (\ln \left (2\right )\right )-18 x \ln \left (2\right )-18 \ln \left (\ln \left (2\right )\right ) x -216}{3 \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )}}\) | \(50\) |
orering | \(-\frac {4 x \left (3 x -14\right ) \left (\left (-6 x +54\right ) {\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}+2 x^{2}-3\right ) {\mathrm e}^{-\frac {x^{2}}{3}}}{12 x^{3}-110 x^{2}+18 x +159}-\frac {9 \left (-1+6 x \right ) \left (\frac {\left (-6 \,{\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}+6 \left (-6 x +54\right ) {\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}+4 x \right ) {\mathrm e}^{-\frac {x^{2}}{3}}}{3}-\frac {2 \left (\left (-6 x +54\right ) {\mathrm e}^{\frac {6 x \ln \left (2 \ln \left (2\right )\right )+72}{\ln \left (2 \ln \left (2\right )\right )}}+2 x^{2}-3\right ) {\mathrm e}^{-\frac {x^{2}}{3}} x}{9}\right )}{2 \left (12 x^{3}-110 x^{2}+18 x +159\right )}\) | \(196\) |
Input:
int(1/3*((-6*x+54)*exp((6*x*ln(2*ln(2))+72)/ln(2*ln(2)))+2*x^2-3)/exp(1/3* x^2),x,method=_RETURNVERBOSE)
Output:
-x*exp(-1/3*x^2)+3*exp(-1/3*x^2+6*x+72/ln(2*ln(2)))
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=-x e^{\left (-\frac {1}{3} \, x^{2}\right )} + 3 \, e^{\left (-\frac {1}{3} \, x^{2} + \frac {6 \, {\left (x \log \left (2 \, \log \left (2\right )\right ) + 12\right )}}{\log \left (2 \, \log \left (2\right )\right )}\right )} \] Input:
integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3 )/exp(1/3*x^2),x, algorithm="fricas")
Output:
-x*e^(-1/3*x^2) + 3*e^(-1/3*x^2 + 6*(x*log(2*log(2)) + 12)/log(2*log(2)))
Time = 0.43 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=- x e^{- \frac {x^{2}}{3}} + 3 e^{- \frac {x^{2}}{3}} e^{\frac {6 x \log {\left (2 \log {\left (2 \right )} \right )} + 72}{\log {\left (2 \log {\left (2 \right )} \right )}}} \] Input:
integrate(1/3*((-6*x+54)*exp((6*x*ln(2*ln(2))+72)/ln(2*ln(2)))+2*x**2-3)/e xp(1/3*x**2),x)
Output:
-x*exp(-x**2/3) + 3*exp(-x**2/3)*exp((6*x*log(2*log(2)) + 72)/log(2*log(2) ))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.55 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=9 \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{3} \, \sqrt {3} x - 3 \, \sqrt {3}\right ) e^{\left (\frac {72}{\log \left (2\right ) + \log \left (\log \left (2\right )\right )} + 27\right )} - i \, \sqrt {3} {\left (-\frac {9 i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (x - 9\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{3}} \sqrt {{\left (x - 9\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x - 9\right )}^{2}}} + i \, \sqrt {3} e^{\left (-\frac {1}{3} \, {\left (x - 9\right )}^{2}\right )}\right )} e^{\left (\frac {72}{\log \left (2 \, \log \left (2\right )\right )} + 27\right )} - x e^{\left (-\frac {1}{3} \, x^{2}\right )} \] Input:
integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3 )/exp(1/3*x^2),x, algorithm="maxima")
Output:
9*sqrt(3)*sqrt(pi)*erf(1/3*sqrt(3)*x - 3*sqrt(3))*e^(72/(log(2) + log(log( 2))) + 27) - I*sqrt(3)*(-9*I*sqrt(3)*sqrt(1/3)*sqrt(pi)*(x - 9)*(erf(sqrt( 1/3)*sqrt((x - 9)^2)) - 1)/sqrt((x - 9)^2) + I*sqrt(3)*e^(-1/3*(x - 9)^2)) *e^(72/log(2*log(2)) + 27) - x*e^(-1/3*x^2)
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=-x e^{\left (-\frac {1}{3} \, x^{2}\right )} + 3 \, e^{\left (-\frac {x^{2} \log \left (2 \, \log \left (2\right )\right ) - 18 \, x \log \left (2 \, \log \left (2\right )\right ) - 216}{3 \, \log \left (2 \, \log \left (2\right )\right )}\right )} \] Input:
integrate(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3 )/exp(1/3*x^2),x, algorithm="giac")
Output:
-x*e^(-1/3*x^2) + 3*e^(-1/3*(x^2*log(2*log(2)) - 18*x*log(2*log(2)) - 216) /log(2*log(2)))
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=3\,{64}^{\frac {x}{\ln \left (2\,\ln \left (2\right )\right )}}\,{\mathrm {e}}^{\frac {72}{\ln \left (2\,\ln \left (2\right )\right )}}\,{\mathrm {e}}^{-\frac {x^2}{3}}\,{\ln \left (2\right )}^{\frac {6\,x}{\ln \left (\ln \left (4\right )\right )}}-x\,{\mathrm {e}}^{-\frac {x^2}{3}} \] Input:
int(-exp(-x^2/3)*((exp((6*x*log(2*log(2)) + 72)/log(2*log(2)))*(6*x - 54)) /3 - (2*x^2)/3 + 1),x)
Output:
3*64^(x/log(2*log(2)))*exp(72/log(2*log(2)))*exp(-x^2/3)*log(2)^((6*x)/log (log(4))) - x*exp(-x^2/3)
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{3} e^{-\frac {x^2}{3}} \left (-3+e^{\frac {72+6 x \log (\log (4))}{\log (\log (4))}} (54-6 x)+2 x^2\right ) \, dx=\frac {3 e^{\frac {6 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x +72}{\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )}}-x}{e^{\frac {x^{2}}{3}}} \] Input:
int(1/3*((-6*x+54)*exp((6*x*log(2*log(2))+72)/log(2*log(2)))+2*x^2-3)/exp( 1/3*x^2),x)
Output:
(3*e**((6*log(2*log(2))*x + 72)/log(2*log(2))) - x)/e**(x**2/3)