\(\int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+(-18 e^{3+x}-6 e^{x^2} x) \log (\log (3))+9 \log ^2(\log (3))} \, dx\) [9046]

Optimal result

Integrand size = 89, antiderivative size = 26 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {3 x}{e^{3+x}+\frac {e^{x^2} x}{3}-\log (\log (3))} \]

[Out]

3*x/(exp(3+x)+1/3*exp(x^2)*x-ln(ln(3)))

Rubi [F]

\[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx \]

[In]

Int[(E^(3 + x)*(27 - 27*x) - 18*E^x^2*x^3 - 27*Log[Log[3]])/(9*E^(6 + 2*x) + 6*E^(3 + x + x^2)*x + E^(2*x^2)*x
^2 + (-18*E^(3 + x) - 6*E^x^2*x)*Log[Log[3]] + 9*Log[Log[3]]^2),x]

[Out]

-27*Log[Log[3]]*Defer[Int][(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]])^(-2), x] + 27*Defer[Int][E^(3 + x)/(3*E^(3
+ x) + E^x^2*x - 3*Log[Log[3]])^2, x] - 27*Defer[Int][(E^(3 + x)*x)/(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]])^2,
x] - 54*Log[Log[3]]*Defer[Int][x^2/(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]])^2, x] + 54*Defer[Int][(E^(3 + x)*x
^2)/(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]])^2, x] - 18*Defer[Int][x^2/(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]]),
x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {9 \left (-3 e^{3+x} (-1+x)-2 e^{x^2} x^3-3 \log (\log (3))\right )}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx \\ & = 9 \int \frac {-3 e^{3+x} (-1+x)-2 e^{x^2} x^3-3 \log (\log (3))}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx \\ & = 9 \int \left (-\frac {2 x^2}{3 e^{3+x}+e^{x^2} x-3 \log (\log (3))}+\frac {3 \left (e^{3+x}-e^{3+x} x+2 e^{3+x} x^2-\log (\log (3))-2 x^2 \log (\log (3))\right )}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}\right ) \, dx \\ & = -\left (18 \int \frac {x^2}{3 e^{3+x}+e^{x^2} x-3 \log (\log (3))} \, dx\right )+27 \int \frac {e^{3+x}-e^{3+x} x+2 e^{3+x} x^2-\log (\log (3))-2 x^2 \log (\log (3))}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx \\ & = -\left (18 \int \frac {x^2}{3 e^{3+x}+e^{x^2} x-3 \log (\log (3))} \, dx\right )+27 \int \left (\frac {e^{3+x}}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}-\frac {e^{3+x} x}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}+\frac {2 e^{3+x} x^2}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}-\frac {\log (\log (3))}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}-\frac {2 x^2 \log (\log (3))}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2}\right ) \, dx \\ & = -\left (18 \int \frac {x^2}{3 e^{3+x}+e^{x^2} x-3 \log (\log (3))} \, dx\right )+27 \int \frac {e^{3+x}}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx-27 \int \frac {e^{3+x} x}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx+54 \int \frac {e^{3+x} x^2}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx-(27 \log (\log (3))) \int \frac {1}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx-(54 \log (\log (3))) \int \frac {x^2}{\left (3 e^{3+x}+e^{x^2} x-3 \log (\log (3))\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {9 x}{3 e^{3+x}+e^{x^2} x-3 \log (\log (3))} \]

[In]

Integrate[(E^(3 + x)*(27 - 27*x) - 18*E^x^2*x^3 - 27*Log[Log[3]])/(9*E^(6 + 2*x) + 6*E^(3 + x + x^2)*x + E^(2*
x^2)*x^2 + (-18*E^(3 + x) - 6*E^x^2*x)*Log[Log[3]] + 9*Log[Log[3]]^2),x]

[Out]

(9*x)/(3*E^(3 + x) + E^x^2*x - 3*Log[Log[3]])

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {9 x}{-{\mathrm e}^{x^{2}} x +3 \ln \left (\ln \left (3\right )\right )-3 \,{\mathrm e}^{3+x}}\)	\(25\)
parallelrisch	\(-\frac {9 x}{-{\mathrm e}^{x^{2}} x +3 \ln \left (\ln \left (3\right )\right )-3 \,{\mathrm e}^{3+x}}\)	\(25\)

[In]

int((-27*ln(ln(3))-18*x^3*exp(x^2)+(-27*x+27)*exp(3+x))/(9*ln(ln(3))^2+(-6*exp(x^2)*x-18*exp(3+x))*ln(ln(3))+x
^2*exp(x^2)^2+6*x*exp(3+x)*exp(x^2)+9*exp(3+x)^2),x,method=_RETURNVERBOSE)

[Out]

-9*x/(-exp(x^2)*x+3*ln(ln(3))-3*exp(3+x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {9 \, x e^{\left (x^{2}\right )}}{x e^{\left (2 \, x^{2}\right )} - 3 \, e^{\left (x^{2}\right )} \log \left (\log \left (3\right )\right ) + 3 \, e^{\left (x^{2} + x + 3\right )}} \]

[In]

integrate((-27*log(log(3))-18*x^3*exp(x^2)+(-27*x+27)*exp(3+x))/(9*log(log(3))^2+(-6*exp(x^2)*x-18*exp(3+x))*l
og(log(3))+x^2*exp(x^2)^2+6*x*exp(3+x)*exp(x^2)+9*exp(3+x)^2),x, algorithm="fricas")

[Out]

9*x*e^(x^2)/(x*e^(2*x^2) - 3*e^(x^2)*log(log(3)) + 3*e^(x^2 + x + 3))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {3 x}{\frac {x e^{x^{2}}}{3} + e^{x + 3} - \log {\left (\log {\left (3 \right )} \right )}} \]

[In]

integrate((-27*ln(ln(3))-18*x**3*exp(x**2)+(-27*x+27)*exp(3+x))/(9*ln(ln(3))**2+(-6*exp(x**2)*x-18*exp(3+x))*l
n(ln(3))+x**2*exp(x**2)**2+6*x*exp(3+x)*exp(x**2)+9*exp(3+x)**2),x)

[Out]

3*x/(x*exp(x**2)/3 + exp(x + 3) - log(log(3)))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {9 \, x}{x e^{\left (x^{2}\right )} + 3 \, e^{\left (x + 3\right )} - 3 \, \log \left (\log \left (3\right )\right )} \]

[In]

integrate((-27*log(log(3))-18*x^3*exp(x^2)+(-27*x+27)*exp(3+x))/(9*log(log(3))^2+(-6*exp(x^2)*x-18*exp(3+x))*l
og(log(3))+x^2*exp(x^2)^2+6*x*exp(3+x)*exp(x^2)+9*exp(3+x)^2),x, algorithm="maxima")

[Out]

9*x/(x*e^(x^2) + 3*e^(x + 3) - 3*log(log(3)))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\frac {9 \, x}{x e^{\left (x^{2}\right )} + 3 \, e^{\left (x + 3\right )} - 3 \, \log \left (\log \left (3\right )\right )} \]

[In]

integrate((-27*log(log(3))-18*x^3*exp(x^2)+(-27*x+27)*exp(3+x))/(9*log(log(3))^2+(-6*exp(x^2)*x-18*exp(3+x))*l
og(log(3))+x^2*exp(x^2)^2+6*x*exp(3+x)*exp(x^2)+9*exp(3+x)^2),x, algorithm="giac")

[Out]

9*x/(x*e^(x^2) + 3*e^(x + 3) - 3*log(log(3)))

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{3+x} (27-27 x)-18 e^{x^2} x^3-27 \log (\log (3))}{9 e^{6+2 x}+6 e^{3+x+x^2} x+e^{2 x^2} x^2+\left (-18 e^{3+x}-6 e^{x^2} x\right ) \log (\log (3))+9 \log ^2(\log (3))} \, dx=\int -\frac {27\,\ln \left (\ln \left (3\right )\right )+18\,x^3\,{\mathrm {e}}^{x^2}+{\mathrm {e}}^{x+3}\,\left (27\,x-27\right )}{9\,{\mathrm {e}}^{2\,x+6}+9\,{\ln \left (\ln \left (3\right )\right )}^2-\ln \left (\ln \left (3\right )\right )\,\left (18\,{\mathrm {e}}^{x+3}+6\,x\,{\mathrm {e}}^{x^2}\right )+x^2\,{\mathrm {e}}^{2\,x^2}+6\,x\,{\mathrm {e}}^{x+3}\,{\mathrm {e}}^{x^2}} \,d x \]

[In]

int(-(27*log(log(3)) + 18*x^3*exp(x^2) + exp(x + 3)*(27*x - 27))/(9*exp(2*x + 6) + 9*log(log(3))^2 - log(log(3
))*(18*exp(x + 3) + 6*x*exp(x^2)) + x^2*exp(2*x^2) + 6*x*exp(x + 3)*exp(x^2)),x)

[Out]

int(-(27*log(log(3)) + 18*x^3*exp(x^2) + exp(x + 3)*(27*x - 27))/(9*exp(2*x + 6) + 9*log(log(3))^2 - log(log(3
))*(18*exp(x + 3) + 6*x*exp(x^2)) + x^2*exp(2*x^2) + 6*x*exp(x + 3)*exp(x^2)), x)