Integrand size = 123, antiderivative size = 29 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{-x^2 \left (e^{3-e^{2 x}}+\log (\log (4))\right )^2}}{x} \] Output:
4/x/exp((ln(2*ln(2))+exp(-exp(x)^2+3))^2*x^2)
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{-x^2 \left (e^{6-2 e^{2 x}}+\log ^2(\log (4))\right )} \log ^{-2 e^{3-e^{2 x}} x^2}(4)}{x} \] Input:
Integrate[(E^(-(E^(6 - 2*E^(2*x))*x^2) - 2*E^(3 - E^(2*x))*x^2*Log[Log[4]] - x^2*Log[Log[4]]^2)*(-4 + E^(6 - 2*E^(2*x))*(-8*x^2 + 16*E^(2*x)*x^3) + E^(3 - E^(2*x))*(-16*x^2 + 16*E^(2*x)*x^3)*Log[Log[4]] - 8*x^2*Log[Log[4]] ^2))/x^2,x]
Output:
4/(E^(x^2*(E^(6 - 2*E^(2*x)) + Log[Log[4]]^2))*x*Log[4]^(2*E^(3 - E^(2*x)) *x^2))
Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(29)=58\).
Time = 1.39 (sec) , antiderivative size = 200, normalized size of antiderivative = 6.90, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {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.
| \(\displaystyle \int \frac {\left (-8 x^2 \log ^2(\log (4))+e^{6-2 e^{2 x}} \left (16 e^{2 x} x^3-8 x^2\right )+e^{3-e^{2 x}} \left (16 e^{2 x} x^3-16 x^2\right ) \log (\log (4))-4\right ) \exp \left (-e^{6-2 e^{2 x}} x^2-x^2 \log ^2(\log (4))-2 e^{3-e^{2 x}} x^2 \log (\log (4))\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {4 e^{-e^{6-2 e^{2 x}} x^2-x^2 \log ^2(\log (4))} \log ^{-2 e^{3-e^{2 x}} x^2}(4) \left (x^2 \log ^2(\log (4))+e^{6-2 e^{2 x}} \left (x^2-2 e^{2 x} x^3\right )+2 e^{3-e^{2 x}} \left (x^2-e^{2 x} x^3\right ) \log (\log (4))\right )}{x^2 \left (-2 e^{2 x-2 e^{2 x}+6} x^2-2 e^{2 x-e^{2 x}+3} x^2 \log (\log (4))+e^{6-2 e^{2 x}} x+x \log ^2(\log (4))+2 e^{3-e^{2 x}} x \log (\log (4))\right )}\) |
Input:
Int[(E^(-(E^(6 - 2*E^(2*x))*x^2) - 2*E^(3 - E^(2*x))*x^2*Log[Log[4]] - x^2 *Log[Log[4]]^2)*(-4 + E^(6 - 2*E^(2*x))*(-8*x^2 + 16*E^(2*x)*x^3) + E^(3 - E^(2*x))*(-16*x^2 + 16*E^(2*x)*x^3)*Log[Log[4]] - 8*x^2*Log[Log[4]]^2))/x ^2,x]
Output:
(4*E^(-(E^(6 - 2*E^(2*x))*x^2) - x^2*Log[Log[4]]^2)*(E^(6 - 2*E^(2*x))*(x^ 2 - 2*E^(2*x)*x^3) + 2*E^(3 - E^(2*x))*(x^2 - E^(2*x)*x^3)*Log[Log[4]] + x ^2*Log[Log[4]]^2))/(x^2*Log[4]^(2*E^(3 - E^(2*x))*x^2)*(E^(6 - 2*E^(2*x))* x - 2*E^(6 - 2*E^(2*x) + 2*x)*x^2 + 2*E^(3 - E^(2*x))*x*Log[Log[4]] - 2*E^ (3 - E^(2*x) + 2*x)*x^2*Log[Log[4]] + x*Log[Log[4]]^2))
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 = 3.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66
| method | result | size |
| parallelrisch | \(\frac {4 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{-{\mathrm e}^{2 x}+3} \ln \left (4 \ln \left (2\right )^{2}\right )+\ln \left (2 \ln \left (2\right )\right )^{2}+{\mathrm e}^{-2 \,{\mathrm e}^{2 x}+6}\right )}}{x}\) | \(48\) |
| risch | \(\frac {4 \,4^{-x^{2} {\mathrm e}^{-{\mathrm e}^{2 x}+3}} \ln \left (2\right )^{-2 x^{2} {\mathrm e}^{-{\mathrm e}^{2 x}+3}} \ln \left (2\right )^{-2 x^{2} \ln \left (2\right )} {\mathrm e}^{-x^{2} \left (\ln \left (\ln \left (2\right )\right )^{2}+\ln \left (2\right )^{2}+{\mathrm e}^{-2 \,{\mathrm e}^{2 x}+6}\right )}}{x}\) | \(79\) |
Input:
int(((16*exp(x)^2*x^3-8*x^2)*exp(-exp(x)^2+3)^2+(16*exp(x)^2*x^3-16*x^2)*l n(2*ln(2))*exp(-exp(x)^2+3)-8*x^2*ln(2*ln(2))^2-4)/x^2/exp(x^2*exp(-exp(x) ^2+3)^2+2*x^2*ln(2*ln(2))*exp(-exp(x)^2+3)+x^2*ln(2*ln(2))^2),x,method=_RE TURNVERBOSE)
Output:
4/exp(x^2*(exp(-exp(x)^2+3)^2+2*ln(2*ln(2))*exp(-exp(x)^2+3)+ln(2*ln(2))^2 ))/x
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 \, e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x} \] Input:
integrate(((16*exp(x)^2*x^3-8*x^2)*exp(-exp(x)^2+3)^2+(16*exp(x)^2*x^3-16* x^2)*log(2*log(2))*exp(-exp(x)^2+3)-8*x^2*log(2*log(2))^2-4)/x^2/exp(x^2*e xp(-exp(x)^2+3)^2+2*x^2*log(2*log(2))*exp(-exp(x)^2+3)+x^2*log(2*log(2))^2 ),x, algorithm="fricas")
Output:
4*e^(-2*x^2*e^(-e^(2*x) + 3)*log(2*log(2)) - x^2*log(2*log(2))^2 - x^2*e^( -2*e^(2*x) + 6))/x
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{- 2 x^{2} e^{3 - e^{2 x}} \log {\left (2 \log {\left (2 \right )} \right )} - x^{2} e^{6 - 2 e^{2 x}} - x^{2} \log {\left (2 \log {\left (2 \right )} \right )}^{2}}}{x} \] Input:
integrate(((16*exp(x)**2*x**3-8*x**2)*exp(-exp(x)**2+3)**2+(16*exp(x)**2*x **3-16*x**2)*ln(2*ln(2))*exp(-exp(x)**2+3)-8*x**2*ln(2*ln(2))**2-4)/x**2/e xp(x**2*exp(-exp(x)**2+3)**2+2*x**2*ln(2*ln(2))*exp(-exp(x)**2+3)+x**2*ln( 2*ln(2))**2),x)
Output:
4*exp(-2*x**2*exp(3 - exp(2*x))*log(2*log(2)) - x**2*exp(6 - 2*exp(2*x)) - x**2*log(2*log(2))**2)/x
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 \, e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2\right ) - x^{2} \log \left (2\right )^{2} - 2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (\log \left (2\right )\right ) - 2 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) - x^{2} \log \left (\log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x} \] Input:
integrate(((16*exp(x)^2*x^3-8*x^2)*exp(-exp(x)^2+3)^2+(16*exp(x)^2*x^3-16* x^2)*log(2*log(2))*exp(-exp(x)^2+3)-8*x^2*log(2*log(2))^2-4)/x^2/exp(x^2*e xp(-exp(x)^2+3)^2+2*x^2*log(2*log(2))*exp(-exp(x)^2+3)+x^2*log(2*log(2))^2 ),x, algorithm="maxima")
Output:
4*e^(-2*x^2*e^(-e^(2*x) + 3)*log(2) - x^2*log(2)^2 - 2*x^2*e^(-e^(2*x) + 3 )*log(log(2)) - 2*x^2*log(2)*log(log(2)) - x^2*log(log(2))^2 - x^2*e^(-2*e ^(2*x) + 6))/x
\[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\int { -\frac {4 \, {\left (2 \, x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - 4 \, {\left (x^{3} e^{\left (2 \, x\right )} - x^{2}\right )} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - 2 \, {\left (2 \, x^{3} e^{\left (2 \, x\right )} - x^{2}\right )} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )} + 1\right )} e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x^{2}} \,d x } \] Input:
integrate(((16*exp(x)^2*x^3-8*x^2)*exp(-exp(x)^2+3)^2+(16*exp(x)^2*x^3-16* x^2)*log(2*log(2))*exp(-exp(x)^2+3)-8*x^2*log(2*log(2))^2-4)/x^2/exp(x^2*e xp(-exp(x)^2+3)^2+2*x^2*log(2*log(2))*exp(-exp(x)^2+3)+x^2*log(2*log(2))^2 ),x, algorithm="giac")
Output:
undef
Time = 7.53 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{-x^2\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-x^2\,{\ln \left (\ln \left (2\right )\right )}^2}}{2^{2\,x^2\,\ln \left (\ln \left (2\right )\right )}\,2^{2\,x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^3}\,x\,{\ln \left (2\right )}^{2\,x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^3}} \] Input:
int((exp(- x^2*exp(6 - 2*exp(2*x)) - x^2*log(2*log(2))^2 - 2*x^2*log(2*log (2))*exp(3 - exp(2*x)))*(exp(6 - 2*exp(2*x))*(16*x^3*exp(2*x) - 8*x^2) - 8 *x^2*log(2*log(2))^2 + log(2*log(2))*exp(3 - exp(2*x))*(16*x^3*exp(2*x) - 16*x^2) - 4))/x^2,x)
Output:
(4*exp(-x^2*exp(-2*exp(2*x))*exp(6))*exp(-x^2*log(2)^2)*exp(-x^2*log(log(2 ))^2))/(2^(2*x^2*log(log(2)))*2^(2*x^2*exp(-exp(2*x))*exp(3))*x*log(2)^(2* x^2*exp(-exp(2*x))*exp(3)))
\[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=16 \left (\int \frac {e^{2 x} x}{e^{\frac {e^{2 e^{2 x}+2 x}+e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}}}d x \right ) \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3}+16 \left (\int \frac {e^{2 x} x}{e^{\frac {2 e^{2 e^{2 x}+2 x}+e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}}}d x \right ) e^{6}-16 \left (\int \frac {1}{e^{\frac {e^{2 e^{2 x}+2 x}+e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}}}d x \right ) \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3}-8 \left (\int \frac {1}{e^{\frac {e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}}}d x \right ) \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2}-8 \left (\int \frac {1}{e^{\frac {2 e^{2 e^{2 x}+2 x}+e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}}}d x \right ) e^{6}-4 \left (\int \frac {1}{e^{\frac {e^{2 e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2} x^{2}+2 e^{e^{2 x}} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{3} x^{2}+e^{6} x^{2}}{e^{2 e^{2 x}}}} x^{2}}d x \right ) \] Input:
int(((16*exp(x)^2*x^3-8*x^2)*exp(-exp(x)^2+3)^2+(16*exp(x)^2*x^3-16*x^2)*l og(2*log(2))*exp(-exp(x)^2+3)-8*x^2*log(2*log(2))^2-4)/x^2/exp(x^2*exp(-ex p(x)^2+3)^2+2*x^2*log(2*log(2))*exp(-exp(x)^2+3)+x^2*log(2*log(2))^2),x)
Output:
4*(4*int((e**(2*x)*x)/e**((e**(2*e**(2*x) + 2*x) + e**(2*e**(2*x))*log(2*l og(2))**2*x**2 + 2*e**(e**(2*x))*log(2*log(2))*e**3*x**2 + e**6*x**2)/e**( 2*e**(2*x))),x)*log(2*log(2))*e**3 + 4*int((e**(2*x)*x)/e**((2*e**(2*e**(2 *x) + 2*x) + e**(2*e**(2*x))*log(2*log(2))**2*x**2 + 2*e**(e**(2*x))*log(2 *log(2))*e**3*x**2 + e**6*x**2)/e**(2*e**(2*x))),x)*e**6 - 4*int(1/e**((e* *(2*e**(2*x) + 2*x) + e**(2*e**(2*x))*log(2*log(2))**2*x**2 + 2*e**(e**(2* x))*log(2*log(2))*e**3*x**2 + e**6*x**2)/e**(2*e**(2*x))),x)*log(2*log(2)) *e**3 - 2*int(1/e**((e**(2*e**(2*x))*log(2*log(2))**2*x**2 + 2*e**(e**(2*x ))*log(2*log(2))*e**3*x**2 + e**6*x**2)/e**(2*e**(2*x))),x)*log(2*log(2))* *2 - 2*int(1/e**((2*e**(2*e**(2*x) + 2*x) + e**(2*e**(2*x))*log(2*log(2))* *2*x**2 + 2*e**(e**(2*x))*log(2*log(2))*e**3*x**2 + e**6*x**2)/e**(2*e**(2 *x))),x)*e**6 - int(1/(e**((e**(2*e**(2*x))*log(2*log(2))**2*x**2 + 2*e**( e**(2*x))*log(2*log(2))*e**3*x**2 + e**6*x**2)/e**(2*e**(2*x)))*x**2),x))