Integrand size = 180, antiderivative size = 23 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\frac {e^{e^{x^2}}}{\left (-\frac {5 x}{-1+e^3}+\log (x)\right )^2} \end {dmath*}
Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\frac {e^{e^{x^2}} \left (-1+e^3\right )^2}{\left (5 x+\log (x)-e^3 \log (x)\right )^2} \end {dmath*}
Integrate[(E^E^x^2*(2 - 2*E^9 + E^3*(-6 - 20*x) + 10*x + E^6*(6 + 10*x) + E^x^2*(-10*x^3 + 20*E^3*x^3 - 10*E^6*x^3) + E^x^2*(-2*x^2 + 6*E^3*x^2 - 6* E^6*x^2 + 2*E^9*x^2)*Log[x]))/(-125*x^4 + (-75*x^3 + 75*E^3*x^3)*Log[x] + (-15*x^2 + 30*E^3*x^2 - 15*E^6*x^2)*Log[x]^2 + (-x + 3*E^3*x - 3*E^6*x + E ^9*x)*Log[x]^3),x]
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(23)=46\).
Time = 0.91 (sec) , antiderivative size = 144, normalized size of antiderivative = 6.26, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, 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.
\(\displaystyle \int \frac {e^{e^{x^2}} \left (e^{x^2} \left (2 e^9 x^2-6 e^6 x^2+6 e^3 x^2-2 x^2\right ) \log (x)+e^{x^2} \left (-10 e^6 x^3+20 e^3 x^3-10 x^3\right )+e^3 (-20 x-6)+10 x+e^6 (10 x+6)-2 e^9+2\right )}{-125 x^4+\left (75 e^3 x^3-75 x^3\right ) \log (x)+\left (-15 e^6 x^2+30 e^3 x^2-15 x^2\right ) \log ^2(x)+\left (e^9 x-3 e^6 x+3 e^3 x-x\right ) \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {e^{e^{x^2}-x^2} \left (e^{x^2} \left (-e^9 x^2+3 e^6 x^2-3 e^3 x^2+x^2\right ) \log (x)+5 e^{x^2} \left (e^6 x^3-2 e^3 x^3+x^3\right )\right )}{x \left (125 x^4+75 \left (x^3-e^3 x^3\right ) \log (x)+15 \left (e^6 x^2-2 e^3 x^2+x^2\right ) \log ^2(x)+\left (1-e^3\right )^3 x \log ^3(x)\right )}\) |
Int[(E^E^x^2*(2 - 2*E^9 + E^3*(-6 - 20*x) + 10*x + E^6*(6 + 10*x) + E^x^2* (-10*x^3 + 20*E^3*x^3 - 10*E^6*x^3) + E^x^2*(-2*x^2 + 6*E^3*x^2 - 6*E^6*x^ 2 + 2*E^9*x^2)*Log[x]))/(-125*x^4 + (-75*x^3 + 75*E^3*x^3)*Log[x] + (-15*x ^2 + 30*E^3*x^2 - 15*E^6*x^2)*Log[x]^2 + (-x + 3*E^3*x - 3*E^6*x + E^9*x)* Log[x]^3),x]
(E^(E^x^2 - x^2)*(5*E^x^2*(x^3 - 2*E^3*x^3 + E^6*x^3) + E^x^2*(x^2 - 3*E^3 *x^2 + 3*E^6*x^2 - E^9*x^2)*Log[x]))/(x*(125*x^4 + 75*(x^3 - E^3*x^3)*Log[ x] + 15*(x^2 - 2*E^3*x^2 + E^6*x^2)*Log[x]^2 + (1 - E^3)^3*x*Log[x]^3))
3.6.94.3.1 Defintions of rubi rules used
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 = 6.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {\left (-2 \,{\mathrm e}^{3}+1+{\mathrm e}^{6}\right ) {\mathrm e}^{{\mathrm e}^{x^{2}}}}{\left (\ln \left (x \right ) {\mathrm e}^{3}-\ln \left (x \right )-5 x \right )^{2}}\) | \(30\) |
parallelrisch | \(\frac {25 \,{\mathrm e}^{6} {\mathrm e}^{{\mathrm e}^{x^{2}}}-50 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x^{2}}}+25 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}}{25 \,{\mathrm e}^{6} \ln \left (x \right )^{2}-250 x \,{\mathrm e}^{3} \ln \left (x \right )-50 \,{\mathrm e}^{3} \ln \left (x \right )^{2}+625 x^{2}+250 x \ln \left (x \right )+25 \ln \left (x \right )^{2}}\) | \(72\) |
int(((2*x^2*exp(3)^3-6*x^2*exp(3)^2+6*x^2*exp(3)-2*x^2)*exp(x^2)*ln(x)+(-1 0*x^3*exp(3)^2+20*x^3*exp(3)-10*x^3)*exp(x^2)-2*exp(3)^3+(10*x+6)*exp(3)^2 +(-20*x-6)*exp(3)+10*x+2)*exp(exp(x^2))/((x*exp(3)^3-3*x*exp(3)^2+3*x*exp( 3)-x)*ln(x)^3+(-15*x^2*exp(3)^2+30*x^2*exp(3)-15*x^2)*ln(x)^2+(75*x^3*exp( 3)-75*x^3)*ln(x)-125*x^4),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\frac {{\left (e^{6} - 2 \, e^{3} + 1\right )} e^{\left (e^{\left (x^{2}\right )}\right )}}{{\left (e^{6} - 2 \, e^{3} + 1\right )} \log \left (x\right )^{2} + 25 \, x^{2} - 10 \, {\left (x e^{3} - x\right )} \log \left (x\right )} \end {dmath*}
integrate(((2*x^2*exp(3)^3-6*x^2*exp(3)^2+6*x^2*exp(3)-2*x^2)*exp(x^2)*log (x)+(-10*x^3*exp(3)^2+20*x^3*exp(3)-10*x^3)*exp(x^2)-2*exp(3)^3+(10*x+6)*e xp(3)^2+(-20*x-6)*exp(3)+10*x+2)*exp(exp(x^2))/((x*exp(3)^3-3*x*exp(3)^2+3 *x*exp(3)-x)*log(x)^3+(-15*x^2*exp(3)^2+30*x^2*exp(3)-15*x^2)*log(x)^2+(75 *x^3*exp(3)-75*x^3)*log(x)-125*x^4),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\frac {\left (- 2 e^{3} + 1 + e^{6}\right ) e^{e^{x^{2}}}}{25 x^{2} - 10 x e^{3} \log {\left (x \right )} + 10 x \log {\left (x \right )} - 2 e^{3} \log {\left (x \right )}^{2} + \log {\left (x \right )}^{2} + e^{6} \log {\left (x \right )}^{2}} \end {dmath*}
integrate(((2*x**2*exp(3)**3-6*x**2*exp(3)**2+6*x**2*exp(3)-2*x**2)*exp(x* *2)*ln(x)+(-10*x**3*exp(3)**2+20*x**3*exp(3)-10*x**3)*exp(x**2)-2*exp(3)** 3+(10*x+6)*exp(3)**2+(-20*x-6)*exp(3)+10*x+2)*exp(exp(x**2))/((x*exp(3)**3 -3*x*exp(3)**2+3*x*exp(3)-x)*ln(x)**3+(-15*x**2*exp(3)**2+30*x**2*exp(3)-1 5*x**2)*ln(x)**2+(75*x**3*exp(3)-75*x**3)*ln(x)-125*x**4),x)
(-2*exp(3) + 1 + exp(6))*exp(exp(x**2))/(25*x**2 - 10*x*exp(3)*log(x) + 10 *x*log(x) - 2*exp(3)*log(x)**2 + log(x)**2 + exp(6)*log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=-\frac {{\left (e^{6} - 2 \, e^{3} + 1\right )} e^{\left (e^{\left (x^{2}\right )}\right )}}{10 \, x {\left (e^{3} - 1\right )} \log \left (x\right ) - {\left (e^{6} - 2 \, e^{3} + 1\right )} \log \left (x\right )^{2} - 25 \, x^{2}} \end {dmath*}
integrate(((2*x^2*exp(3)^3-6*x^2*exp(3)^2+6*x^2*exp(3)-2*x^2)*exp(x^2)*log (x)+(-10*x^3*exp(3)^2+20*x^3*exp(3)-10*x^3)*exp(x^2)-2*exp(3)^3+(10*x+6)*e xp(3)^2+(-20*x-6)*exp(3)+10*x+2)*exp(exp(x^2))/((x*exp(3)^3-3*x*exp(3)^2+3 *x*exp(3)-x)*log(x)^3+(-15*x^2*exp(3)^2+30*x^2*exp(3)-15*x^2)*log(x)^2+(75 *x^3*exp(3)-75*x^3)*log(x)-125*x^4),x, algorithm=\
-(e^6 - 2*e^3 + 1)*e^(e^(x^2))/(10*x*(e^3 - 1)*log(x) - (e^6 - 2*e^3 + 1)* log(x)^2 - 25*x^2)
\begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\int { -\frac {2 \, {\left ({\left (x^{2} e^{9} - 3 \, x^{2} e^{6} + 3 \, x^{2} e^{3} - x^{2}\right )} e^{\left (x^{2}\right )} \log \left (x\right ) + {\left (5 \, x + 3\right )} e^{6} - {\left (10 \, x + 3\right )} e^{3} - 5 \, {\left (x^{3} e^{6} - 2 \, x^{3} e^{3} + x^{3}\right )} e^{\left (x^{2}\right )} + 5 \, x - e^{9} + 1\right )} e^{\left (e^{\left (x^{2}\right )}\right )}}{125 \, x^{4} - {\left (x e^{9} - 3 \, x e^{6} + 3 \, x e^{3} - x\right )} \log \left (x\right )^{3} + 15 \, {\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2}\right )} \log \left (x\right )^{2} - 75 \, {\left (x^{3} e^{3} - x^{3}\right )} \log \left (x\right )} \,d x } \end {dmath*}
integrate(((2*x^2*exp(3)^3-6*x^2*exp(3)^2+6*x^2*exp(3)-2*x^2)*exp(x^2)*log (x)+(-10*x^3*exp(3)^2+20*x^3*exp(3)-10*x^3)*exp(x^2)-2*exp(3)^3+(10*x+6)*e xp(3)^2+(-20*x-6)*exp(3)+10*x+2)*exp(exp(x^2))/((x*exp(3)^3-3*x*exp(3)^2+3 *x*exp(3)-x)*log(x)^3+(-15*x^2*exp(3)^2+30*x^2*exp(3)-15*x^2)*log(x)^2+(75 *x^3*exp(3)-75*x^3)*log(x)-125*x^4),x, algorithm=\
integrate(-2*((x^2*e^9 - 3*x^2*e^6 + 3*x^2*e^3 - x^2)*e^(x^2)*log(x) + (5* x + 3)*e^6 - (10*x + 3)*e^3 - 5*(x^3*e^6 - 2*x^3*e^3 + x^3)*e^(x^2) + 5*x - e^9 + 1)*e^(e^(x^2))/(125*x^4 - (x*e^9 - 3*x*e^6 + 3*x*e^3 - x)*log(x)^3 + 15*(x^2*e^6 - 2*x^2*e^3 + x^2)*log(x)^2 - 75*(x^3*e^3 - x^3)*log(x)), x )
Time = 14.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \begin {dmath*} \int \frac {e^{e^{x^2}} \left (2-2 e^9+e^3 (-6-20 x)+10 x+e^6 (6+10 x)+e^{x^2} \left (-10 x^3+20 e^3 x^3-10 e^6 x^3\right )+e^{x^2} \left (-2 x^2+6 e^3 x^2-6 e^6 x^2+2 e^9 x^2\right ) \log (x)\right )}{-125 x^4+\left (-75 x^3+75 e^3 x^3\right ) \log (x)+\left (-15 x^2+30 e^3 x^2-15 e^6 x^2\right ) \log ^2(x)+\left (-x+3 e^3 x-3 e^6 x+e^9 x\right ) \log ^3(x)} \, dx=\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^2}}\,{\left ({\mathrm {e}}^3-1\right )}^2}{{\left (5\,x-\ln \left (x\right )\,\left ({\mathrm {e}}^3-1\right )\right )}^2} \end {dmath*}
int(-(exp(exp(x^2))*(10*x - 2*exp(9) - exp(x^2)*(10*x^3*exp(6) - 20*x^3*ex p(3) + 10*x^3) + exp(6)*(10*x + 6) - exp(3)*(20*x + 6) + exp(x^2)*log(x)*( 6*x^2*exp(3) - 6*x^2*exp(6) + 2*x^2*exp(9) - 2*x^2) + 2))/(log(x)^3*(x - 3 *x*exp(3) + 3*x*exp(6) - x*exp(9)) + log(x)^2*(15*x^2*exp(6) - 30*x^2*exp( 3) + 15*x^2) - log(x)*(75*x^3*exp(3) - 75*x^3) + 125*x^4),x)