\(\int \frac {-13 e^{x/2}-52 e^{x^2} x+(-78+26 e^{x/2}+26 e^{x^2}) \log (-3+e^{x/2}+e^{x^2})}{(-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2) \log (-3+e^{x/2}+e^{x^2})+(72 x-24 e^{x/2} x-24 e^{x^2} x) \log (-3+e^{x/2}+e^{x^2}) \log (\log (-3+e^{x/2}+e^{x^2}))+(-36+12 e^{x/2}+12 e^{x^2}) \log (-3+e^{x/2}+e^{x^2}) \log ^2(\log (-3+e^{x/2}+e^{x^2}))} \, dx\) [1746]

Optimal result

Integrand size = 206, antiderivative size = 26 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{6 \left (-x+\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )} \] Output:

13/6/(ln(ln(exp(x^2)+exp(1/2*x)-3))-x)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{6 \left (-x+\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )} \] Input:

Integrate[(-13*E^(x/2) - 52*E^x^2*x + (-78 + 26*E^(x/2) + 26*E^x^2)*Log[-3 
 + E^(x/2) + E^x^2])/((-36*x^2 + 12*E^(x/2)*x^2 + 12*E^x^2*x^2)*Log[-3 + E 
^(x/2) + E^x^2] + (72*x - 24*E^(x/2)*x - 24*E^x^2*x)*Log[-3 + E^(x/2) + E^ 
x^2]*Log[Log[-3 + E^(x/2) + E^x^2]] + (-36 + 12*E^(x/2) + 12*E^x^2)*Log[-3 
 + E^(x/2) + E^x^2]*Log[Log[-3 + E^(x/2) + E^x^2]]^2),x]
 

Output:

13/(6*(-x + Log[Log[-3 + E^(x/2) + E^x^2]]))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {7239, 27, 7237}

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.

\(\displaystyle \int \frac {-52 e^{x^2} x+\left (26 e^{x^2}+26 e^{x/2}-78\right ) \log \left (e^{x^2}+e^{x/2}-3\right )-13 e^{x/2}}{\left (12 e^{x^2}+12 e^{x/2}-36\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \log ^2\left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )+\left (-24 e^{x^2} x-24 e^{x/2} x+72 x\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )+\left (12 e^{x^2} x^2+12 e^{x/2} x^2-36 x^2\right ) \log \left (e^{x^2}+e^{x/2}-3\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {13 \left (4 e^{x^2} x-2 \left (e^{x^2}+e^{x/2}-3\right ) \log \left (e^{x^2}+e^{x/2}-3\right )+e^{x/2}\right )}{12 \left (-e^{x^2}-e^{x/2}+3\right ) \log \left (e^{x^2}+e^{x/2}-3\right ) \left (x-\log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13}{12} \int \frac {4 e^{x^2} x+e^{x/2}+2 \left (3-e^{x/2}-e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (3-e^{x/2}-e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \left (x-\log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )\right )^2}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle -\frac {13}{6 \left (x-\log \left (\log \left (e^{x^2}+e^{x/2}-3\right )\right )\right )}\)

Input:

Int[(-13*E^(x/2) - 52*E^x^2*x + (-78 + 26*E^(x/2) + 26*E^x^2)*Log[-3 + E^( 
x/2) + E^x^2])/((-36*x^2 + 12*E^(x/2)*x^2 + 12*E^x^2*x^2)*Log[-3 + E^(x/2) 
 + E^x^2] + (72*x - 24*E^(x/2)*x - 24*E^x^2*x)*Log[-3 + E^(x/2) + E^x^2]*L 
og[Log[-3 + E^(x/2) + E^x^2]] + (-36 + 12*E^(x/2) + 12*E^x^2)*Log[-3 + E^( 
x/2) + E^x^2]*Log[Log[-3 + E^(x/2) + E^x^2]]^2),x]
 

Output:

-13/(6*(x - Log[Log[-3 + E^(x/2) + E^x^2]]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [A] (verified)

Time = 11.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {13}{6 \left (x -\ln \left (\ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )\right )}\)	\(21\)
parallelrisch	\(-\frac {13}{6 \left (x -\ln \left (\ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )\right )}\)	\(21\)

Input:

int(((26*exp(x^2)+26*exp(1/2*x)-78)*ln(exp(x^2)+exp(1/2*x)-3)-52*exp(x^2)* 
x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*ln(exp(x^2)+exp(1/2*x)-3) 
*ln(ln(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x)+72*x)*ln( 
exp(x^2)+exp(1/2*x)-3)*ln(ln(exp(x^2)+exp(1/2*x)-3))+(12*x^2*exp(x^2)+12*x 
^2*exp(1/2*x)-36*x^2)*ln(exp(x^2)+exp(1/2*x)-3)),x,method=_RETURNVERBOSE)
 

Output:

-13/6/(x-ln(ln(exp(x^2)+exp(1/2*x)-3)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \] Input:

integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex 
p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 
/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) 
+72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*exp( 
x^2)*x^2+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith 
m="fricas")
 

Output:

-13/6/(x - log(log(e^(x^2) + e^(1/2*x) - 3)))
 

Sympy [A] (verification not implemented)

Time = 6.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\frac {13}{- 6 x + 6 \log {\left (\log {\left (e^{\frac {x}{2}} + e^{x^{2}} - 3 \right )} \right )}} \] Input:

integrate(((26*exp(x**2)+26*exp(1/2*x)-78)*ln(exp(x**2)+exp(1/2*x)-3)-52*e 
xp(x**2)*x-13*exp(1/2*x))/((12*exp(x**2)+12*exp(1/2*x)-36)*ln(exp(x**2)+ex 
p(1/2*x)-3)*ln(ln(exp(x**2)+exp(1/2*x)-3))**2+(-24*exp(x**2)*x-24*x*exp(1/ 
2*x)+72*x)*ln(exp(x**2)+exp(1/2*x)-3)*ln(ln(exp(x**2)+exp(1/2*x)-3))+(12*e 
xp(x**2)*x**2+12*x**2*exp(1/2*x)-36*x**2)*ln(exp(x**2)+exp(1/2*x)-3)),x)
 

Output:

13/(-6*x + 6*log(log(exp(x/2) + exp(x**2) - 3)))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \] Input:

integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex 
p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 
/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) 
+72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*exp( 
x^2)*x^2+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith 
m="maxima")
 

Output:

-13/6/(x - log(log(e^(x^2) + e^(1/2*x) - 3)))
 

Giac [A] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6 \, {\left (x - \log \left (\log \left (e^{\left (x^{2}\right )} + e^{\left (\frac {1}{2} \, x\right )} - 3\right )\right )\right )}} \] Input:

integrate(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*ex 
p(x^2)*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1 
/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x) 
+72*x)*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*exp( 
x^2)*x^2+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x, algorith 
m="giac")
 

Output:

-13/6/(x - log(log(e^(x^2) + e^(1/2*x) - 3)))
 

Mupad [B] (verification not implemented)

Time = 3.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=-\frac {13}{6\,\left (x-\ln \left (\ln \left ({\mathrm {e}}^{x/2}+{\mathrm {e}}^{x^2}-3\right )\right )\right )} \] Input:

int(-(13*exp(x/2) + 52*x*exp(x^2) - log(exp(x/2) + exp(x^2) - 3)*(26*exp(x 
/2) + 26*exp(x^2) - 78))/(log(exp(x/2) + exp(x^2) - 3)*(12*x^2*exp(x/2) + 
12*x^2*exp(x^2) - 36*x^2) + log(exp(x/2) + exp(x^2) - 3)*log(log(exp(x/2) 
+ exp(x^2) - 3))^2*(12*exp(x/2) + 12*exp(x^2) - 36) - log(exp(x/2) + exp(x 
^2) - 3)*log(log(exp(x/2) + exp(x^2) - 3))*(24*x*exp(x/2) - 72*x + 24*x*ex 
p(x^2))),x)
 

Output:

-13/(6*(x - log(log(exp(x/2) + exp(x^2) - 3))))
 

Reduce [F]

\[ \int \frac {-13 e^{x/2}-52 e^{x^2} x+\left (-78+26 e^{x/2}+26 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )}{\left (-36 x^2+12 e^{x/2} x^2+12 e^{x^2} x^2\right ) \log \left (-3+e^{x/2}+e^{x^2}\right )+\left (72 x-24 e^{x/2} x-24 e^{x^2} x\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log \left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )+\left (-36+12 e^{x/2}+12 e^{x^2}\right ) \log \left (-3+e^{x/2}+e^{x^2}\right ) \log ^2\left (\log \left (-3+e^{x/2}+e^{x^2}\right )\right )} \, dx=\int \frac {\left (26 \,{\mathrm e}^{x^{2}}+26 \,{\mathrm e}^{\frac {x}{2}}-78\right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )-52 \,{\mathrm e}^{x^{2}} x -13 \,{\mathrm e}^{\frac {x}{2}}}{\left (12 \,{\mathrm e}^{x^{2}}+12 \,{\mathrm e}^{\frac {x}{2}}-36\right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right ) {\mathrm {log}\left (\mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )}^{2}+\left (-24 \,{\mathrm e}^{x^{2}} x -24 x \,{\mathrm e}^{\frac {x}{2}}+72 x \right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right ) \mathrm {log}\left (\mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )\right )+\left (12 \,{\mathrm e}^{x^{2}} x^{2}+12 x^{2} {\mathrm e}^{\frac {x}{2}}-36 x^{2}\right ) \mathrm {log}\left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\frac {x}{2}}-3\right )}d x \] Input:

int(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*exp(x^2) 
*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1/2*x)- 
3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x)+72*x) 
*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*exp(x^2)*x 
^2+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x)
 

Output:

int(((26*exp(x^2)+26*exp(1/2*x)-78)*log(exp(x^2)+exp(1/2*x)-3)-52*exp(x^2) 
*x-13*exp(1/2*x))/((12*exp(x^2)+12*exp(1/2*x)-36)*log(exp(x^2)+exp(1/2*x)- 
3)*log(log(exp(x^2)+exp(1/2*x)-3))^2+(-24*exp(x^2)*x-24*x*exp(1/2*x)+72*x) 
*log(exp(x^2)+exp(1/2*x)-3)*log(log(exp(x^2)+exp(1/2*x)-3))+(12*exp(x^2)*x 
^2+12*x^2*exp(1/2*x)-36*x^2)*log(exp(x^2)+exp(1/2*x)-3)),x)