\(\int e^{1-e^{18 x^7 \log ^8(e^5 x)}+2 e^{9 x^7 \log ^8(e^5 x)} x-x^2} (-2 x+e^{18 x^7 \log ^8(e^5 x)} (-144 x^6 \log ^7(e^5 x)-126 x^6 \log ^8(e^5 x))+e^{9 x^7 \log ^8(e^5 x)} (2+144 x^7 \log ^7(e^5 x)+126 x^7 \log ^8(e^5 x))) \, dx\) [2189]

Optimal result

Integrand size = 136, antiderivative size = 27 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=e^{1-\left (e^{9 x^7 \log ^8\left (e^5 x\right )}-x\right )^2} \] Output:

exp(1-(exp(9*x^7*ln(x*exp(5))^8)-x)^2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(27)=54\).

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.70 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=e^{1-x^2-e^{18 x^7 \left (390625+437500 \log ^2(x)+175000 \log ^3(x)+43750 \log ^4(x)+7000 \log ^5(x)+700 \log ^6(x)+40 \log ^7(x)+\log ^8(x)\right )} x^{11250000 x^7}+2 e^{9 x^7 \left (390625+437500 \log ^2(x)+175000 \log ^3(x)+43750 \log ^4(x)+7000 \log ^5(x)+700 \log ^6(x)+40 \log ^7(x)+\log ^8(x)\right )} x^{1+5625000 x^7}} \] Input:

Integrate[E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^ 
2)*(-2*x + E^(18*x^7*Log[E^5*x]^8)*(-144*x^6*Log[E^5*x]^7 - 126*x^6*Log[E^ 
5*x]^8) + E^(9*x^7*Log[E^5*x]^8)*(2 + 144*x^7*Log[E^5*x]^7 + 126*x^7*Log[E 
^5*x]^8)),x]
 

Output:

E^(1 - x^2 - E^(18*x^7*(390625 + 437500*Log[x]^2 + 175000*Log[x]^3 + 43750 
*Log[x]^4 + 7000*Log[x]^5 + 700*Log[x]^6 + 40*Log[x]^7 + Log[x]^8))*x^(112 
50000*x^7) + 2*E^(9*x^7*(390625 + 437500*Log[x]^2 + 175000*Log[x]^3 + 4375 
0*Log[x]^4 + 7000*Log[x]^5 + 700*Log[x]^6 + 40*Log[x]^7 + Log[x]^8))*x^(1 
+ 5625000*x^7))
 

Rubi [A] (verified)

Time = 5.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {7257}

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.

\(\displaystyle \int \left (e^{9 x^7 \log ^8\left (e^5 x\right )} \left (126 x^7 \log ^8\left (e^5 x\right )+144 x^7 \log ^7\left (e^5 x\right )+2\right )+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-126 x^6 \log ^8\left (e^5 x\right )-144 x^6 \log ^7\left (e^5 x\right )\right )-2 x\right ) \exp \left (2 x e^{9 x^7 \log ^8\left (e^5 x\right )}-e^{18 x^7 \log ^8\left (e^5 x\right )}-x^2+1\right ) \, dx\)

\(\Big \downarrow \) 7257

\(\displaystyle \exp \left (2 x e^{9 x^7 \log ^8\left (e^5 x\right )}-e^{18 x^7 \log ^8\left (e^5 x\right )}-x^2+1\right )\)

Input:

Int[E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^2)*(-2 
*x + E^(18*x^7*Log[E^5*x]^8)*(-144*x^6*Log[E^5*x]^7 - 126*x^6*Log[E^5*x]^8 
) + E^(9*x^7*Log[E^5*x]^8)*(2 + 144*x^7*Log[E^5*x]^7 + 126*x^7*Log[E^5*x]^ 
8)),x]
 

Output:

E^(1 - E^(18*x^7*Log[E^5*x]^8) + 2*E^(9*x^7*Log[E^5*x]^8)*x - x^2)
 

Defintions of rubi rules used

rule 7257

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim 
p[q*(F^v/Log[F]), x] /;  !FalseQ[q]] /; FreeQ[F, x]
 

Maple [A] (verified)

Time = 40.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48

method	result	size
risch	\({\mathrm e}^{-{\mathrm e}^{18 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}+2 x \,{\mathrm e}^{9 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}-x^{2}+1}\)	\(40\)
parallelrisch	\({\mathrm e}^{-{\mathrm e}^{18 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}+2 x \,{\mathrm e}^{9 x^{7} \ln \left (x \,{\mathrm e}^{5}\right )^{8}}-x^{2}+1}\)	\(42\)

Input:

int(((-126*x^6*ln(x*exp(5))^8-144*x^6*ln(x*exp(5))^7)*exp(9*x^7*ln(x*exp(5 
))^8)^2+(126*x^7*ln(x*exp(5))^8+144*x^7*ln(x*exp(5))^7+2)*exp(9*x^7*ln(x*e 
xp(5))^8)-2*x)*exp(-exp(9*x^7*ln(x*exp(5))^8)^2+2*x*exp(9*x^7*ln(x*exp(5)) 
^8)-x^2+1),x,method=_RETURNVERBOSE)
 

Output:

exp(-exp(18*x^7*ln(x*exp(5))^8)+2*x*exp(9*x^7*ln(x*exp(5))^8)-x^2+1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=e^{\left (-x^{2} + 2 \, x e^{\left (9 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} - e^{\left (18 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} + 1\right )} \] Input:

integrate(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*lo 
g(x*exp(5))^8)^2+(126*x^7*log(x*exp(5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9 
*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x^7 
*log(x*exp(5))^8)-x^2+1),x, algorithm="fricas")
 

Output:

e^(-x^2 + 2*x*e^(9*x^7*log(x*e^5)^8) - e^(18*x^7*log(x*e^5)^8) + 1)
 

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=e^{- x^{2} + 2 x e^{9 x^{7} \log {\left (x e^{5} \right )}^{8}} - e^{18 x^{7} \log {\left (x e^{5} \right )}^{8}} + 1} \] Input:

integrate(((-126*x**6*ln(x*exp(5))**8-144*x**6*ln(x*exp(5))**7)*exp(9*x**7 
*ln(x*exp(5))**8)**2+(126*x**7*ln(x*exp(5))**8+144*x**7*ln(x*exp(5))**7+2) 
*exp(9*x**7*ln(x*exp(5))**8)-2*x)*exp(-exp(9*x**7*ln(x*exp(5))**8)**2+2*x* 
exp(9*x**7*ln(x*exp(5))**8)-x**2+1),x)
 

Output:

exp(-x**2 + 2*x*exp(9*x**7*log(x*exp(5))**8) - exp(18*x**7*log(x*exp(5))** 
8) + 1)
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*lo 
g(x*exp(5))^8)^2+(126*x^7*log(x*exp(5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9 
*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x^7 
*log(x*exp(5))^8)-x^2+1),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=\int { -2 \, {\left (9 \, {\left (7 \, x^{6} \log \left (x e^{5}\right )^{8} + 8 \, x^{6} \log \left (x e^{5}\right )^{7}\right )} e^{\left (18 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} - {\left (63 \, x^{7} \log \left (x e^{5}\right )^{8} + 72 \, x^{7} \log \left (x e^{5}\right )^{7} + 1\right )} e^{\left (9 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} + x\right )} e^{\left (-x^{2} + 2 \, x e^{\left (9 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} - e^{\left (18 \, x^{7} \log \left (x e^{5}\right )^{8}\right )} + 1\right )} \,d x } \] Input:

integrate(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*lo 
g(x*exp(5))^8)^2+(126*x^7*log(x*exp(5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9 
*x^7*log(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x^7 
*log(x*exp(5))^8)-x^2+1),x, algorithm="giac")
 

Output:

integrate(-2*(9*(7*x^6*log(x*e^5)^8 + 8*x^6*log(x*e^5)^7)*e^(18*x^7*log(x* 
e^5)^8) - (63*x^7*log(x*e^5)^8 + 72*x^7*log(x*e^5)^7 + 1)*e^(9*x^7*log(x*e 
^5)^8) + x)*e^(-x^2 + 2*x*e^(9*x^7*log(x*e^5)^8) - e^(18*x^7*log(x*e^5)^8) 
 + 1), x)
 

Mupad [B] (verification not implemented)

Time = 4.68 (sec) , antiderivative size = 182, normalized size of antiderivative = 6.74 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx={\mathrm {e}}^{2\,x\,x^{5625000\,x^7}\,{\mathrm {e}}^{3515625\,x^7}\,{\mathrm {e}}^{9\,x^7\,{\ln \left (x\right )}^8}\,{\mathrm {e}}^{360\,x^7\,{\ln \left (x\right )}^7}\,{\mathrm {e}}^{6300\,x^7\,{\ln \left (x\right )}^6}\,{\mathrm {e}}^{63000\,x^7\,{\ln \left (x\right )}^5}\,{\mathrm {e}}^{393750\,x^7\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{1575000\,x^7\,{\ln \left (x\right )}^3}\,{\mathrm {e}}^{3937500\,x^7\,{\ln \left (x\right )}^2}}\,\mathrm {e}\,{\mathrm {e}}^{-x^{11250000\,x^7}\,{\mathrm {e}}^{7031250\,x^7}\,{\mathrm {e}}^{18\,x^7\,{\ln \left (x\right )}^8}\,{\mathrm {e}}^{720\,x^7\,{\ln \left (x\right )}^7}\,{\mathrm {e}}^{12600\,x^7\,{\ln \left (x\right )}^6}\,{\mathrm {e}}^{126000\,x^7\,{\ln \left (x\right )}^5}\,{\mathrm {e}}^{787500\,x^7\,{\ln \left (x\right )}^4}\,{\mathrm {e}}^{3150000\,x^7\,{\ln \left (x\right )}^3}\,{\mathrm {e}}^{7875000\,x^7\,{\ln \left (x\right )}^2}}\,{\mathrm {e}}^{-x^2} \] Input:

int(-exp(2*x*exp(9*x^7*log(x*exp(5))^8) - exp(18*x^7*log(x*exp(5))^8) - x^ 
2 + 1)*(2*x - exp(9*x^7*log(x*exp(5))^8)*(144*x^7*log(x*exp(5))^7 + 126*x^ 
7*log(x*exp(5))^8 + 2) + exp(18*x^7*log(x*exp(5))^8)*(144*x^6*log(x*exp(5) 
)^7 + 126*x^6*log(x*exp(5))^8)),x)
 

Output:

exp(2*x*x^(5625000*x^7)*exp(3515625*x^7)*exp(9*x^7*log(x)^8)*exp(360*x^7*l 
og(x)^7)*exp(6300*x^7*log(x)^6)*exp(63000*x^7*log(x)^5)*exp(393750*x^7*log 
(x)^4)*exp(1575000*x^7*log(x)^3)*exp(3937500*x^7*log(x)^2))*exp(1)*exp(-x^ 
(11250000*x^7)*exp(7031250*x^7)*exp(18*x^7*log(x)^8)*exp(720*x^7*log(x)^7) 
*exp(12600*x^7*log(x)^6)*exp(126000*x^7*log(x)^5)*exp(787500*x^7*log(x)^4) 
*exp(3150000*x^7*log(x)^3)*exp(7875000*x^7*log(x)^2))*exp(-x^2)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int e^{1-e^{18 x^7 \log ^8\left (e^5 x\right )}+2 e^{9 x^7 \log ^8\left (e^5 x\right )} x-x^2} \left (-2 x+e^{18 x^7 \log ^8\left (e^5 x\right )} \left (-144 x^6 \log ^7\left (e^5 x\right )-126 x^6 \log ^8\left (e^5 x\right )\right )+e^{9 x^7 \log ^8\left (e^5 x\right )} \left (2+144 x^7 \log ^7\left (e^5 x\right )+126 x^7 \log ^8\left (e^5 x\right )\right )\right ) \, dx=\frac {e^{2 e^{9 \mathrm {log}\left (e^{5} x \right )^{8} x^{7}} x} e}{e^{e^{18 \mathrm {log}\left (e^{5} x \right )^{8} x^{7}}+x^{2}}} \] Input:

int(((-126*x^6*log(x*exp(5))^8-144*x^6*log(x*exp(5))^7)*exp(9*x^7*log(x*ex 
p(5))^8)^2+(126*x^7*log(x*exp(5))^8+144*x^7*log(x*exp(5))^7+2)*exp(9*x^7*l 
og(x*exp(5))^8)-2*x)*exp(-exp(9*x^7*log(x*exp(5))^8)^2+2*x*exp(9*x^7*log(x 
*exp(5))^8)-x^2+1),x)
 

Output:

(e**(2*e**(9*log(e**5*x)**8*x**7)*x)*e)/e**(e**(18*log(e**5*x)**8*x**7) + 
x**2)