Integrand size = 77, antiderivative size = 25 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{e^{1+\frac {1}{2} (-4-x+18 \log (3))^2}}}{x} \] Output:
exp(exp(1+1/2*(18*ln(3)-4-x)^2))/x
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{3^{-72-18 x} e^{9+4 x+\frac {x^2}{2}+162 \log ^2(3)}}}{x} \] Input:
Integrate[(E^E^((18 + 8*x + x^2 + (-144 - 36*x)*Log[3] + 324*Log[3]^2)/2)* (-1 + E^((18 + 8*x + x^2 + (-144 - 36*x)*Log[3] + 324*Log[3]^2)/2)*(4*x + x^2 - 18*x*Log[3])))/x^2,x]
Output:
E^(3^(-72 - 18*x)*E^(9 + 4*x + x^2/2 + 162*Log[3]^2))/x
Leaf count is larger than twice the leaf count of optimal. \(124\) vs. \(2(25)=50\).
Time = 0.77 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.96, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, 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.
| \(\displaystyle \int \frac {\exp \left (\exp \left (\frac {1}{2} \left (x^2+8 x+(-36 x-144) \log (3)+18+324 \log ^2(3)\right )\right )\right ) \left (\left (x^2+4 x-18 x \log (3)\right ) \exp \left (\frac {1}{2} \left (x^2+8 x+(-36 x-144) \log (3)+18+324 \log ^2(3)\right )\right )-1\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {\left (x^2+4 x-18 x \log (3)\right ) \exp \left (\frac {1}{2} \left (-x^2-8 x+36 (x+4) \log (3)-18 \left (1+18 \log ^2(3)\right )\right )+3^{\frac {1}{2} (-36 x-144)} e^{\frac {1}{2} \left (x^2+8 x+18 \left (1+18 \log ^2(3)\right )\right )}+\frac {1}{2} \left (x^2+8 x-36 (x+4) \log (3)+18 \left (1+18 \log ^2(3)\right )\right )\right )}{x^2 (x+2 (2-9 \log (3)))}\) |
Input:
Int[(E^E^((18 + 8*x + x^2 + (-144 - 36*x)*Log[3] + 324*Log[3]^2)/2)*(-1 + E^((18 + 8*x + x^2 + (-144 - 36*x)*Log[3] + 324*Log[3]^2)/2)*(4*x + x^2 - 18*x*Log[3])))/x^2,x]
Output:
(E^(3^((-144 - 36*x)/2)*E^((8*x + x^2 + 18*(1 + 18*Log[3]^2))/2) + (-8*x - x^2 + 36*(4 + x)*Log[3] - 18*(1 + 18*Log[3]^2))/2 + (8*x + x^2 - 36*(4 + x)*Log[3] + 18*(1 + 18*Log[3]^2))/2)*(4*x + x^2 - 18*x*Log[3]))/(x^2*(x + 2*(2 - 9*Log[3])))
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 = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {{\mathrm e}^{3^{-72-18 x} {\mathrm e}^{162 \ln \left (3\right )^{2}+9+\frac {x^{2}}{2}+4 x}}}{x}\) | \(31\) |
| norman | \(\frac {{\mathrm e}^{{\mathrm e}^{162 \ln \left (3\right )^{2}+\frac {\left (-36 x -144\right ) \ln \left (3\right )}{2}+\frac {x^{2}}{2}+4 x +9}}}{x}\) | \(32\) |
Input:
int(((-18*x*ln(3)+x^2+4*x)*exp(162*ln(3)^2+1/2*(-36*x-144)*ln(3)+1/2*x^2+4 *x+9)-1)*exp(exp(162*ln(3)^2+1/2*(-36*x-144)*ln(3)+1/2*x^2+4*x+9))/x^2,x,m ethod=_RETURNVERBOSE)
Output:
exp(3^(-72-18*x)*exp(162*ln(3)^2+9+1/2*x^2+4*x))/x
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{\left (e^{\left (\frac {1}{2} \, x^{2} - 18 \, {\left (x + 4\right )} \log \left (3\right ) + 162 \, \log \left (3\right )^{2} + 4 \, x + 9\right )}\right )}}{x} \] Input:
integrate(((-18*x*log(3)+x^2+4*x)*exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+ 1/2*x^2+4*x+9)-1)*exp(exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+1/2*x^2+4*x+ 9))/x^2,x, algorithm="fricas")
Output:
e^(e^(1/2*x^2 - 18*(x + 4)*log(3) + 162*log(3)^2 + 4*x + 9))/x
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{e^{\frac {x^{2}}{2} + 4 x + \left (- 18 x - 72\right ) \log {\left (3 \right )} + 9 + 162 \log {\left (3 \right )}^{2}}}}{x} \] Input:
integrate(((-18*x*ln(3)+x**2+4*x)*exp(162*ln(3)**2+1/2*(-36*x-144)*ln(3)+1 /2*x**2+4*x+9)-1)*exp(exp(162*ln(3)**2+1/2*(-36*x-144)*ln(3)+1/2*x**2+4*x+ 9))/x**2,x)
Output:
exp(exp(x**2/2 + 4*x + (-18*x - 72)*log(3) + 9 + 162*log(3)**2))/x
Time = 0.48 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{\left (\frac {1}{22528399544939174411840147874772641} \, e^{\left (\frac {1}{2} \, x^{2} - 18 \, x \log \left (3\right ) + 162 \, \log \left (3\right )^{2} + 4 \, x + 9\right )}\right )}}{x} \] Input:
integrate(((-18*x*log(3)+x^2+4*x)*exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+ 1/2*x^2+4*x+9)-1)*exp(exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+1/2*x^2+4*x+ 9))/x^2,x, algorithm="maxima")
Output:
e^(1/22528399544939174411840147874772641*e^(1/2*x^2 - 18*x*log(3) + 162*lo g(3)^2 + 4*x + 9))/x
\[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (x^{2} - 18 \, x \log \left (3\right ) + 4 \, x\right )} e^{\left (\frac {1}{2} \, x^{2} - 18 \, {\left (x + 4\right )} \log \left (3\right ) + 162 \, \log \left (3\right )^{2} + 4 \, x + 9\right )} - 1\right )} e^{\left (e^{\left (\frac {1}{2} \, x^{2} - 18 \, {\left (x + 4\right )} \log \left (3\right ) + 162 \, \log \left (3\right )^{2} + 4 \, x + 9\right )}\right )}}{x^{2}} \,d x } \] Input:
integrate(((-18*x*log(3)+x^2+4*x)*exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+ 1/2*x^2+4*x+9)-1)*exp(exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+1/2*x^2+4*x+ 9))/x^2,x, algorithm="giac")
Output:
integrate(((x^2 - 18*x*log(3) + 4*x)*e^(1/2*x^2 - 18*(x + 4)*log(3) + 162* log(3)^2 + 4*x + 9) - 1)*e^(e^(1/2*x^2 - 18*(x + 4)*log(3) + 162*log(3)^2 + 4*x + 9))/x^2, x)
Time = 4.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {{\mathrm {e}}^{\frac {{\left (\frac {1}{387420489}\right )}^x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^9\,{\mathrm {e}}^{162\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{\frac {x^2}{2}}}{22528399544939174411840147874772641}}}{x} \] Input:
int((exp(exp(4*x - (log(3)*(36*x + 144))/2 + 162*log(3)^2 + x^2/2 + 9))*(e xp(4*x - (log(3)*(36*x + 144))/2 + 162*log(3)^2 + x^2/2 + 9)*(4*x - 18*x*l og(3) + x^2) - 1))/x^2,x)
Output:
exp(((1/387420489)^x*exp(4*x)*exp(9)*exp(162*log(3)^2)*exp(x^2/2))/2252839 9544939174411840147874772641)/x
\[ \int \frac {e^{e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )}} \left (-1+e^{\frac {1}{2} \left (18+8 x+x^2+(-144-36 x) \log (3)+324 \log ^2(3)\right )} \left (4 x+x^2-18 x \log (3)\right )\right )}{x^2} \, dx=\frac {e^{162 \mathrm {log}\left (3\right )^{2}} \left (\int \frac {e^{\frac {2 e^{162 \mathrm {log}\left (3\right )^{2}+\frac {x^{2}}{2}+4 x} e^{9}+22528399544939174411840147874772641 \,3^{18 x} x^{2}+180227196359513395294721182998181128 \,3^{18 x} x}{45056799089878348823680295749545282 \,3^{18 x}}}}{3^{18 x}}d x \right ) e^{9}}{22528399544939174411840147874772641}-\frac {2 e^{162 \mathrm {log}\left (3\right )^{2}} \left (\int \frac {e^{\frac {2 e^{162 \mathrm {log}\left (3\right )^{2}+\frac {x^{2}}{2}+4 x} e^{9}+22528399544939174411840147874772641 \,3^{18 x} x^{2}+180227196359513395294721182998181128 \,3^{18 x} x}{45056799089878348823680295749545282 \,3^{18 x}}}}{3^{18 x} x}d x \right ) \mathrm {log}\left (3\right ) e^{9}}{2503155504993241601315571986085849}+\frac {4 e^{162 \mathrm {log}\left (3\right )^{2}} \left (\int \frac {e^{\frac {2 e^{162 \mathrm {log}\left (3\right )^{2}+\frac {x^{2}}{2}+4 x} e^{9}+22528399544939174411840147874772641 \,3^{18 x} x^{2}+180227196359513395294721182998181128 \,3^{18 x} x}{45056799089878348823680295749545282 \,3^{18 x}}}}{3^{18 x} x}d x \right ) e^{9}}{22528399544939174411840147874772641}-\left (\int \frac {e^{\frac {e^{162 \mathrm {log}\left (3\right )^{2}+\frac {x^{2}}{2}+4 x} e^{9}}{22528399544939174411840147874772641 \,3^{18 x}}}}{x^{2}}d x \right ) \] Input:
int(((-18*x*log(3)+x^2+4*x)*exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+1/2*x^ 2+4*x+9)-1)*exp(exp(162*log(3)^2+1/2*(-36*x-144)*log(3)+1/2*x^2+4*x+9))/x^ 2,x)
Output:
(e**(162*log(3)**2)*int(e**((2*e**((324*log(3)**2 + x**2 + 8*x)/2)*e**9 + 22528399544939174411840147874772641*3**(18*x)*x**2 + 180227196359513395294 721182998181128*3**(18*x)*x)/(45056799089878348823680295749545282*3**(18*x )))/3**(18*x),x)*e**9 - 18*e**(162*log(3)**2)*int(e**((2*e**((324*log(3)** 2 + x**2 + 8*x)/2)*e**9 + 22528399544939174411840147874772641*3**(18*x)*x* *2 + 180227196359513395294721182998181128*3**(18*x)*x)/(450567990898783488 23680295749545282*3**(18*x)))/(3**(18*x)*x),x)*log(3)*e**9 + 4*e**(162*log (3)**2)*int(e**((2*e**((324*log(3)**2 + x**2 + 8*x)/2)*e**9 + 225283995449 39174411840147874772641*3**(18*x)*x**2 + 180227196359513395294721182998181 128*3**(18*x)*x)/(45056799089878348823680295749545282*3**(18*x)))/(3**(18* x)*x),x)*e**9 - 22528399544939174411840147874772641*int(e**((e**((324*log( 3)**2 + x**2 + 8*x)/2)*e**9)/(22528399544939174411840147874772641*3**(18*x )))/x**2,x))/22528399544939174411840147874772641