Integrand size = 92, antiderivative size = 27 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=e^{e^5} x \left (-e^2-e^{x^2}+x^{\frac {1}{6+x}}\right ) \]
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=-e^{e^5} x \left (e^2+e^{x^2}-x^{\frac {1}{6+x}}\right ) \]
Integrate[(E^E^5*(E^2*(-36 - 12*x - x^2) + E^x^2*(-36 - 12*x - 73*x^2 - 24 *x^3 - 2*x^4)) + x^(6 + x)^(-1)*(E^E^5*(42 + 13*x + x^2) - E^E^5*x*Log[x]) )/(36 + 12*x + x^2),x]
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.
\(\displaystyle \int \frac {x^{\frac {1}{x+6}} \left (e^{e^5} \left (x^2+13 x+42\right )-e^{e^5} x \log (x)\right )+e^{e^5} \left (e^2 \left (-x^2-12 x-36\right )+e^{x^2} \left (-2 x^4-24 x^3-73 x^2-12 x-36\right )\right )}{x^2+12 x+36} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {x^{\frac {1}{x+6}} \left (e^{e^5} \left (x^2+13 x+42\right )-e^{e^5} x \log (x)\right )+e^{e^5} \left (e^2 \left (-x^2-12 x-36\right )+e^{x^2} \left (-2 x^4-24 x^3-73 x^2-12 x-36\right )\right )}{(x+6)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{e^5} x^{\frac {1}{x+6}} \left (x^2+13 x-x \log (x)+42\right )}{(x+6)^2}-e^{e^5} \left (2 e^{x^2} x^2+e^{x^2}+e^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{e^5} \int x^{\frac {1}{x+6}}dx-e^{e^5} \int \frac {x^{\frac {x+7}{x+6}} \log (x)}{(x+6)^2}dx+\frac {e^{e^5} x^{\frac {1}{x+6}+1} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{x+6},2+\frac {1}{x+6},-\frac {x}{6}\right )}{6 \left (\frac {1}{x+6}+1\right )}-e^{x^2+e^5} x-e^{2+e^5} x\) |
Int[(E^E^5*(E^2*(-36 - 12*x - x^2) + E^x^2*(-36 - 12*x - 73*x^2 - 24*x^3 - 2*x^4)) + x^(6 + x)^(-1)*(E^E^5*(42 + 13*x + x^2) - E^E^5*x*Log[x]))/(36 + 12*x + x^2),x]
3.19.34.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 10.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-x \,{\mathrm e}^{{\mathrm e}^{5}+2}-x \,{\mathrm e}^{x^{2}+{\mathrm e}^{5}}+{\mathrm e}^{{\mathrm e}^{5}} x \,x^{\frac {1}{6+x}}\) | \(32\) |
parallelrisch | \(-{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{2} x -{\mathrm e}^{{\mathrm e}^{5}} x \,{\mathrm e}^{x^{2}}+x \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}+24 \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{2}\) | \(41\) |
default | \(\frac {{\mathrm e}^{{\mathrm e}^{5}} x^{3} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}+36 x \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}+12 x^{2} {\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}}{\left (6+x \right )^{2}}-{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{2} x -{\mathrm e}^{{\mathrm e}^{5}} x \,{\mathrm e}^{x^{2}}\) | \(75\) |
parts | \(\frac {{\mathrm e}^{{\mathrm e}^{5}} x^{3} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}+36 x \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}+12 x^{2} {\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{\frac {\ln \left (x \right )}{6+x}}}{\left (6+x \right )^{2}}-{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{2} x -{\mathrm e}^{{\mathrm e}^{5}} x \,{\mathrm e}^{x^{2}}\) | \(75\) |
int(((-x*exp(exp(5))*ln(x)+(x^2+13*x+42)*exp(exp(5)))*exp(ln(x)/(6+x))+((- 2*x^4-24*x^3-73*x^2-12*x-36)*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(exp(5)))/ (x^2+12*x+36),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=x x^{\left (\frac {1}{x + 6}\right )} e^{\left (e^{5}\right )} - {\left (x e^{2} + x e^{\left (x^{2}\right )}\right )} e^{\left (e^{5}\right )} \]
integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(6 +x))+((-2*x^4-24*x^3-73*x^2-12*x-36)*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(e xp(5)))/(x^2+12*x+36),x, algorithm=\
Time = 106.49 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=- x e^{x^{2}} e^{e^{5}} + x e^{\frac {\log {\left (x \right )}}{x + 6}} e^{e^{5}} - x e^{2} e^{e^{5}} \]
integrate(((-x*exp(exp(5))*ln(x)+(x**2+13*x+42)*exp(exp(5)))*exp(ln(x)/(6+ x))+((-2*x**4-24*x**3-73*x**2-12*x-36)*exp(x**2)+(-x**2-12*x-36)*exp(2))*e xp(exp(5)))/(x**2+12*x+36),x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.89 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=-x e^{\left (x^{2} + e^{5}\right )} + x e^{\left (\frac {\log \left (x\right )}{x + 6} + e^{5}\right )} - {\left (x - \frac {36}{x + 6} - 12 \, \log \left (x + 6\right )\right )} e^{\left (e^{5} + 2\right )} - 12 \, {\left (\frac {6}{x + 6} + \log \left (x + 6\right )\right )} e^{\left (e^{5} + 2\right )} + \frac {36 \, e^{\left (e^{5} + 2\right )}}{x + 6} \]
integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(6 +x))+((-2*x^4-24*x^3-73*x^2-12*x-36)*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(e xp(5)))/(x^2+12*x+36),x, algorithm=\
-x*e^(x^2 + e^5) + x*e^(log(x)/(x + 6) + e^5) - (x - 36/(x + 6) - 12*log(x + 6))*e^(e^5 + 2) - 12*(6/(x + 6) + log(x + 6))*e^(e^5 + 2) + 36*e^(e^5 + 2)/(x + 6)
Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=x^{\frac {x}{x + 6}} x^{\frac {7}{x + 6}} e^{\left (e^{5}\right )} - x e^{\left (x^{2} + e^{5}\right )} - x e^{\left (e^{5} + 2\right )} \]
integrate(((-x*exp(exp(5))*log(x)+(x^2+13*x+42)*exp(exp(5)))*exp(log(x)/(6 +x))+((-2*x^4-24*x^3-73*x^2-12*x-36)*exp(x^2)+(-x^2-12*x-36)*exp(2))*exp(e xp(5)))/(x^2+12*x+36),x, algorithm=\
Time = 11.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {e^{e^5} \left (e^2 \left (-36-12 x-x^2\right )+e^{x^2} \left (-36-12 x-73 x^2-24 x^3-2 x^4\right )\right )+x^{\frac {1}{6+x}} \left (e^{e^5} \left (42+13 x+x^2\right )-e^{e^5} x \log (x)\right )}{36+12 x+x^2} \, dx=-x\,{\mathrm {e}}^{{\mathrm {e}}^5}\,\left ({\mathrm {e}}^{x^2}+{\mathrm {e}}^2-x^{\frac {1}{x+6}}\right ) \]