Integrand size = 39, antiderivative size = 24 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2-e^{-6-x-x^2} (3+x) \log (\log (5)) \] Output:
x^2-(3+x)/exp(x^2+6)/exp(x)*ln(ln(5))
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2+e^{-x-x^2} \left (-\frac {3}{e^6}-\frac {x}{e^6}\right ) \log (\log (5)) \] Input:
Integrate[E^(-6 - x - x^2)*(2*E^(6 + x + x^2)*x + (2 + 7*x + 2*x^2)*Log[Lo g[5]]),x]
Output:
x^2 + E^(-x - x^2)*(-3/E^6 - x/E^6)*Log[Log[5]]
Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {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.
\(\displaystyle \int e^{-x^2-x-6} \left (2 e^{x^2+x+6} x+\left (2 x^2+7 x+2\right ) \log (\log (5))\right ) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (e^{-x^2-x-6} \left (2 x^2+7 x+2\right ) \log (\log (5))+2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2-e^{-x^2-x-6} x \log (\log (5))-3 e^{-x^2-x-6} \log (\log (5))\) |
Input:
Int[E^(-6 - x - x^2)*(2*E^(6 + x + x^2)*x + (2 + 7*x + 2*x^2)*Log[Log[5]]) ,x]
Output:
x^2 - 3*E^(-6 - x - x^2)*Log[Log[5]] - E^(-6 - x - x^2)*x*Log[Log[5]]
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x^{2}-\ln \left (\ln \left (5\right )\right ) \left (3+x \right ) {\mathrm e}^{-x^{2}-x -6}\) | \(24\) |
norman | \(\left (x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}-x \ln \left (\ln \left (5\right )\right )-3 \ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}\) | \(38\) |
parallelrisch | \(-\left (-x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}+x \ln \left (\ln \left (5\right )\right )+3 \ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}\) | \(39\) |
parts | \(-3 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) {\mathrm e}^{-x^{2}-x}-{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) x \,{\mathrm e}^{-x^{2}-x}+x^{2}\) | \(44\) |
default | \(x^{2}+{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )+7 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \left (-\frac {{\mathrm e}^{-x^{2}-x}}{2}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )}{4}\right )+2 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \left (-\frac {x \,{\mathrm e}^{-x^{2}-x}}{2}+\frac {{\mathrm e}^{-x^{2}-x}}{4}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )}{8}\right )\) | \(101\) |
orering | \(\frac {\left (4 x^{5}+16 x^{4}-x^{3}-37 x^{2}-16 x +4\right ) \left (\left (2 x^{2}+7 x +2\right ) \ln \left (\ln \left (5\right )\right )+2 x \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+6}\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}}{8 x^{4}+32 x^{3}+18 x^{2}+4 x +4}+\frac {\left (2 x^{4}+7 x^{3}-8 x -4\right ) \left (\left (\left (4 x +7\right ) \ln \left (\ln \left (5\right )\right )+2 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+6}+2 x \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+6}+4 x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}-\left (\left (2 x^{2}+7 x +2\right ) \ln \left (\ln \left (5\right )\right )+2 x \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+6}\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}-2 \left (\left (2 x^{2}+7 x +2\right ) \ln \left (\ln \left (5\right )\right )+2 x \,{\mathrm e}^{x} {\mathrm e}^{x^{2}+6}\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6} x \right )}{8 x^{4}+32 x^{3}+18 x^{2}+4 x +4}\) | \(267\) |
Input:
int(((2*x^2+7*x+2)*ln(ln(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x,me thod=_RETURNVERBOSE)
Output:
x^2-ln(ln(5))*(3+x)*exp(-x^2-x-6)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx={\left (x^{2} e^{\left (x^{2} + x + 6\right )} - {\left (x + 3\right )} \log \left (\log \left (5\right )\right )\right )} e^{\left (-x^{2} - x - 6\right )} \] Input:
integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2 +6),x, algorithm="fricas")
Output:
(x^2*e^(x^2 + x + 6) - (x + 3)*log(log(5)))*e^(-x^2 - x - 6)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^{2} + \left (- x e^{- x} \log {\left (\log {\left (5 \right )} \right )} - 3 e^{- x} \log {\left (\log {\left (5 \right )} \right )}\right ) e^{- x^{2} - 6} \] Input:
integrate(((2*x**2+7*x+2)*ln(ln(5))+2*x*exp(x)*exp(x**2+6))/exp(x)/exp(x** 2+6),x)
Output:
x**2 + (-x*exp(-x)*log(log(5)) - 3*exp(-x)*log(log(5)))*exp(-x**2 - 6)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=\sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) - \frac {1}{4} i \, {\left (-\frac {4 i \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} + 4 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) - \frac {7}{4} i \, {\left (-\frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} - 2 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) + x^{2} \] Input:
integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2 +6),x, algorithm="maxima")
Output:
sqrt(pi)*erf(x + 1/2)*e^(-23/4)*log(log(5)) - 1/4*I*(-4*I*(2*x + 1)^3*gamm a(3/2, 1/4*(2*x + 1)^2)/((2*x + 1)^2)^(3/2) + I*sqrt(pi)*(2*x + 1)*(erf(1/ 2*sqrt((2*x + 1)^2)) - 1)/sqrt((2*x + 1)^2) + 4*I*e^(-1/4*(2*x + 1)^2))*e^ (-23/4)*log(log(5)) - 7/4*I*(-I*sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt((2*x + 1) ^2)) - 1)/sqrt((2*x + 1)^2) - 2*I*e^(-1/4*(2*x + 1)^2))*e^(-23/4)*log(log( 5)) + x^2
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^{2} - \frac {1}{2} \, {\left ({\left (2 \, x + 1\right )} \log \left (\log \left (5\right )\right ) + 5 \, \log \left (\log \left (5\right )\right )\right )} e^{\left (-x^{2} - x - 6\right )} \] Input:
integrate(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2 +6),x, algorithm="giac")
Output:
x^2 - 1/2*((2*x + 1)*log(log(5)) + 5*log(log(5)))*e^(-x^2 - x - 6)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2-3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \left (5\right )\right )-x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \left (5\right )\right ) \] Input:
int(exp(-x)*exp(- x^2 - 6)*(log(log(5))*(7*x + 2*x^2 + 2) + 2*x*exp(x^2 + 6)*exp(x)),x)
Output:
x^2 - 3*exp(-x)*exp(-6)*exp(-x^2)*log(log(5)) - x*exp(-x)*exp(-6)*exp(-x^2 )*log(log(5))
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=\frac {e^{x^{2}+x} e^{6} x^{2}-\mathrm {log}\left (\mathrm {log}\left (5\right )\right ) x -3 \,\mathrm {log}\left (\mathrm {log}\left (5\right )\right )}{e^{x^{2}+x} e^{6}} \] Input:
int(((2*x^2+7*x+2)*log(log(5))+2*x*exp(x)*exp(x^2+6))/exp(x)/exp(x^2+6),x)
Output:
(e**(x**2 + x)*e**6*x**2 - log(log(5))*x - 3*log(log(5)))/(e**(x**2 + x)*e **6)