\(\int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} (-3 x^2+3 x^3)+e^{2 \log ^2(x)} (9 x^3-9 x^4)+e^{\log ^2(x)} (-12-12 x-9 x^4+9 x^5-48 \log (x))}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} (-8 x+6 x^5)} \, dx\) [905]

Optimal result

Integrand size = 141, antiderivative size = 30 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \left (x-\log \left (x \left (-1+\frac {4}{x^2 \left (-e^{\log ^2(x)}+x\right )^2}\right )\right )\right ) \] Output:

3/2*x-3/2*ln((4/x^2/(x-exp(ln(x)^2))^2-1)*x)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \left (x+2 \log \left (e^{\log ^2(x)}-x\right )+\log (x)-\log \left (2+e^{\log ^2(x)} x-x^2\right )-\log \left (2-e^{\log ^2(x)} x+x^2\right )\right ) \] Input:

Integrate[(36*x + 12*x^2 + 3*x^5 - 3*x^6 + E^(3*Log[x]^2)*(-3*x^2 + 3*x^3) 
 + E^(2*Log[x]^2)*(9*x^3 - 9*x^4) + E^Log[x]^2*(-12 - 12*x - 9*x^4 + 9*x^5 
 - 48*Log[x]))/(8*x^2 + 2*E^(3*Log[x]^2)*x^3 - 6*E^(2*Log[x]^2)*x^4 - 2*x^ 
6 + E^Log[x]^2*(-8*x + 6*x^5)),x]
 

Output:

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

Rubi [F]

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.

\(\displaystyle \int \frac {-3 x^6+3 x^5+12 x^2+e^{\log ^2(x)} \left (9 x^5-9 x^4-12 x-48 \log (x)-12\right )+\left (9 x^3-9 x^4\right ) e^{2 \log ^2(x)}+\left (3 x^3-3 x^2\right ) e^{3 \log ^2(x)}+36 x}{-2 x^6+\left (6 x^5-8 x\right ) e^{\log ^2(x)}-6 x^4 e^{2 \log ^2(x)}+2 x^3 e^{3 \log ^2(x)}+8 x^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {3 x^6-3 x^5-12 x^2-e^{\log ^2(x)} \left (9 x^5-9 x^4-12 x-48 \log (x)-12\right )-\left (9 x^3-9 x^4\right ) e^{2 \log ^2(x)}-\left (3 x^3-3 x^2\right ) e^{3 \log ^2(x)}-36 x}{2 x \left (x^5-3 x^4 e^{\log ^2(x)}+3 x^3 e^{2 \log ^2(x)}-x^2 e^{3 \log ^2(x)}-4 x+4 e^{\log ^2(x)}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int -\frac {3 \left (-x^6+x^5+4 x^2+12 x-e^{3 \log ^2(x)} \left (x^2-x^3\right )+3 e^{2 \log ^2(x)} \left (x^3-x^4\right )-e^{\log ^2(x)} \left (-3 x^5+3 x^4+4 x+16 \log (x)+4\right )\right )}{x \left (x^5-3 e^{\log ^2(x)} x^4+3 e^{2 \log ^2(x)} x^3-e^{3 \log ^2(x)} x^2-4 x+4 e^{\log ^2(x)}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} \int \frac {-x^6+x^5+4 x^2+12 x-e^{3 \log ^2(x)} \left (x^2-x^3\right )+3 e^{2 \log ^2(x)} \left (x^3-x^4\right )-e^{\log ^2(x)} \left (-3 x^5+3 x^4+4 x+16 \log (x)+4\right )}{x \left (x^5-3 e^{\log ^2(x)} x^4+3 e^{2 \log ^2(x)} x^3-e^{3 \log ^2(x)} x^2-4 x+4 e^{\log ^2(x)}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3}{2} \int \left (\frac {1-x}{x}-\frac {2 (2 \log (x)-1)}{e^{\log ^2(x)}-x}-\frac {2 \log (x) x^2-x^2-4 \log (x)-2}{x \left (x^2-e^{\log ^2(x)} x-2\right )}-\frac {2 \log (x) x^2-x^2+4 \log (x)+2}{x \left (x^2-e^{\log ^2(x)} x+2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3}{2} \left (2 \int \frac {1}{x \left (x^2-e^{\log ^2(x)} x-2\right )}dx+\int \frac {x}{x^2-e^{\log ^2(x)} x-2}dx-2 \int \frac {1}{x \left (x^2-e^{\log ^2(x)} x+2\right )}dx+\int \frac {x}{x^2-e^{\log ^2(x)} x+2}dx+4 \int \frac {\log (x)}{x \left (x^2-e^{\log ^2(x)} x-2\right )}dx-2 \int \frac {x \log (x)}{x^2-e^{\log ^2(x)} x-2}dx-4 \int \frac {\log (x)}{x \left (x^2-e^{\log ^2(x)} x+2\right )}dx-2 \int \frac {x \log (x)}{x^2-e^{\log ^2(x)} x+2}dx+2 \int \frac {1}{e^{\log ^2(x)}-x}dx-4 \int \frac {\log (x)}{e^{\log ^2(x)}-x}dx-x+\log (x)\right )\)

Input:

Int[(36*x + 12*x^2 + 3*x^5 - 3*x^6 + E^(3*Log[x]^2)*(-3*x^2 + 3*x^3) + E^( 
2*Log[x]^2)*(9*x^3 - 9*x^4) + E^Log[x]^2*(-12 - 12*x - 9*x^4 + 9*x^5 - 48* 
Log[x]))/(8*x^2 + 2*E^(3*Log[x]^2)*x^3 - 6*E^(2*Log[x]^2)*x^4 - 2*x^6 + E^ 
Log[x]^2*(-8*x + 6*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63

method	result	size
risch	\(\frac {3 x}{2}-\frac {3 \ln \left (x \right )}{2}+3 \ln \left (-x +{\mathrm e}^{\ln \left (x \right )^{2}}\right )-\frac {3 \ln \left ({\mathrm e}^{2 \ln \left (x \right )^{2}}-2 \,{\mathrm e}^{\ln \left (x \right )^{2}} x +\frac {x^{4}-4}{x^{2}}\right )}{2}\)	\(49\)
parallelrisch	\(\frac {3 x}{2}+\frac {3 \ln \left (x \right )}{2}+3 \ln \left (x -{\mathrm e}^{\ln \left (x \right )^{2}}\right )-\frac {3 \ln \left (x^{2}-{\mathrm e}^{\ln \left (x \right )^{2}} x -2\right )}{2}-\frac {3 \ln \left (x^{2}-{\mathrm e}^{\ln \left (x \right )^{2}} x +2\right )}{2}\)	\(53\)

Input:

int(((3*x^3-3*x^2)*exp(ln(x)^2)^3+(-9*x^4+9*x^3)*exp(ln(x)^2)^2+(-48*ln(x) 
+9*x^5-9*x^4-12*x-12)*exp(ln(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp(ln( 
x)^2)^3-6*x^4*exp(ln(x)^2)^2+(6*x^5-8*x)*exp(ln(x)^2)-2*x^6+8*x^2),x,metho 
d=_RETURNVERBOSE)
 

Output:

3/2*x-3/2*ln(x)+3*ln(-x+exp(ln(x)^2))-3/2*ln(exp(2*ln(x)^2)-2*exp(ln(x)^2) 
*x+1/x^2*(x^4-4))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x\right ) + 3 \, \log \left (-x + e^{\left (\log \left (x\right )^{2}\right )}\right ) - \frac {3}{2} \, \log \left (\frac {x^{4} - 2 \, x^{3} e^{\left (\log \left (x\right )^{2}\right )} + x^{2} e^{\left (2 \, \log \left (x\right )^{2}\right )} - 4}{x^{2}}\right ) \] Input:

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(- 
48*log(x)+9*x^5-9*x^4-12*x-12)*exp(log(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x 
^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6+8 
*x^2),x, algorithm="fricas")
 

Output:

3/2*x - 3/2*log(x) + 3*log(-x + e^(log(x)^2)) - 3/2*log((x^4 - 2*x^3*e^(lo 
g(x)^2) + x^2*e^(2*log(x)^2) - 4)/x^2)
 

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3 x}{2} - \frac {3 \log {\left (x \right )}}{2} + 3 \log {\left (- x + e^{\log {\left (x \right )}^{2}} \right )} - \frac {3 \log {\left (- 2 x e^{\log {\left (x \right )}^{2}} + e^{2 \log {\left (x \right )}^{2}} + \frac {x^{4} - 4}{x^{2}} \right )}}{2} \] Input:

integrate(((3*x**3-3*x**2)*exp(ln(x)**2)**3+(-9*x**4+9*x**3)*exp(ln(x)**2) 
**2+(-48*ln(x)+9*x**5-9*x**4-12*x-12)*exp(ln(x)**2)-3*x**6+3*x**5+12*x**2+ 
36*x)/(2*x**3*exp(ln(x)**2)**3-6*x**4*exp(ln(x)**2)**2+(6*x**5-8*x)*exp(ln 
(x)**2)-2*x**6+8*x**2),x)
 

Output:

3*x/2 - 3*log(x)/2 + 3*log(-x + exp(log(x)**2)) - 3*log(-2*x*exp(log(x)**2 
) + exp(2*log(x)**2) + (x**4 - 4)/x**2)/2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x\right ) + 3 \, \log \left (-x + e^{\left (\log \left (x\right )^{2}\right )}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \left (x\right )^{2}\right )} + 2}{x}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \left (x\right )^{2}\right )} - 2}{x}\right ) \] Input:

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(- 
48*log(x)+9*x^5-9*x^4-12*x-12)*exp(log(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x 
^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6+8 
*x^2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 3.82 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x^{4} - 2 \, x^{3} e^{\left (\log \left (x\right )^{2}\right )} + x^{2} e^{\left (2 \, \log \left (x\right )^{2}\right )} - 4\right ) + 3 \, \log \left (x - e^{\left (\log \left (x\right )^{2}\right )}\right ) + \frac {3}{2} \, \log \left (x\right ) \] Input:

integrate(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(- 
48*log(x)+9*x^5-9*x^4-12*x-12)*exp(log(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x 
^3*exp(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6+8 
*x^2),x, algorithm="giac")
 

Output:

3/2*x - 3/2*log(x^4 - 2*x^3*e^(log(x)^2) + x^2*e^(2*log(x)^2) - 4) + 3*log 
(x - e^(log(x)^2)) + 3/2*log(x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\int -\frac {36\,x-{\mathrm {e}}^{3\,{\ln \left (x\right )}^2}\,\left (3\,x^2-3\,x^3\right )+{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,\left (9\,x^3-9\,x^4\right )+12\,x^2+3\,x^5-3\,x^6-{\mathrm {e}}^{{\ln \left (x\right )}^2}\,\left (12\,x+48\,\ln \left (x\right )+9\,x^4-9\,x^5+12\right )}{6\,x^4\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}-2\,x^3\,{\mathrm {e}}^{3\,{\ln \left (x\right )}^2}+{\mathrm {e}}^{{\ln \left (x\right )}^2}\,\left (8\,x-6\,x^5\right )-8\,x^2+2\,x^6} \,d x \] Input:

int(-(36*x - exp(3*log(x)^2)*(3*x^2 - 3*x^3) + exp(2*log(x)^2)*(9*x^3 - 9* 
x^4) + 12*x^2 + 3*x^5 - 3*x^6 - exp(log(x)^2)*(12*x + 48*log(x) + 9*x^4 - 
9*x^5 + 12))/(6*x^4*exp(2*log(x)^2) - 2*x^3*exp(3*log(x)^2) + exp(log(x)^2 
)*(8*x - 6*x^5) - 8*x^2 + 2*x^6),x)
 

Output:

int(-(36*x - exp(3*log(x)^2)*(3*x^2 - 3*x^3) + exp(2*log(x)^2)*(9*x^3 - 9* 
x^4) + 12*x^2 + 3*x^5 - 3*x^6 - exp(log(x)^2)*(12*x + 48*log(x) + 9*x^4 - 
9*x^5 + 12))/(6*x^4*exp(2*log(x)^2) - 2*x^3*exp(3*log(x)^2) + exp(log(x)^2 
)*(8*x - 6*x^5) - 8*x^2 + 2*x^6), x)
 

Reduce [F]

\[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=-12 \left (\int \frac {e^{\mathrm {log}\left (x \right )^{2}}}{e^{3 \mathrm {log}\left (x \right )^{2}} x^{3}-3 e^{2 \mathrm {log}\left (x \right )^{2}} x^{4}+3 e^{\mathrm {log}\left (x \right )^{2}} x^{5}-4 e^{\mathrm {log}\left (x \right )^{2}} x -x^{6}+4 x^{2}}d x \right )-24 \left (\int \frac {e^{\mathrm {log}\left (x \right )^{2}} \mathrm {log}\left (x \right )}{e^{3 \mathrm {log}\left (x \right )^{2}} x^{3}-3 e^{2 \mathrm {log}\left (x \right )^{2}} x^{4}+3 e^{\mathrm {log}\left (x \right )^{2}} x^{5}-4 e^{\mathrm {log}\left (x \right )^{2}} x -x^{6}+4 x^{2}}d x \right )+24 \left (\int \frac {1}{e^{3 \mathrm {log}\left (x \right )^{2}} x^{2}-3 e^{2 \mathrm {log}\left (x \right )^{2}} x^{3}+3 e^{\mathrm {log}\left (x \right )^{2}} x^{4}-4 e^{\mathrm {log}\left (x \right )^{2}}-x^{5}+4 x}d x \right )-\frac {3 \,\mathrm {log}\left (x \right )}{2}+\frac {3 x}{2} \] Input:

int(((3*x^3-3*x^2)*exp(log(x)^2)^3+(-9*x^4+9*x^3)*exp(log(x)^2)^2+(-48*log 
(x)+9*x^5-9*x^4-12*x-12)*exp(log(x)^2)-3*x^6+3*x^5+12*x^2+36*x)/(2*x^3*exp 
(log(x)^2)^3-6*x^4*exp(log(x)^2)^2+(6*x^5-8*x)*exp(log(x)^2)-2*x^6+8*x^2), 
x)
 

Output:

(3*( - 8*int(e**(log(x)**2)/(e**(3*log(x)**2)*x**3 - 3*e**(2*log(x)**2)*x* 
*4 + 3*e**(log(x)**2)*x**5 - 4*e**(log(x)**2)*x - x**6 + 4*x**2),x) - 16*i 
nt((e**(log(x)**2)*log(x))/(e**(3*log(x)**2)*x**3 - 3*e**(2*log(x)**2)*x** 
4 + 3*e**(log(x)**2)*x**5 - 4*e**(log(x)**2)*x - x**6 + 4*x**2),x) + 16*in 
t(1/(e**(3*log(x)**2)*x**2 - 3*e**(2*log(x)**2)*x**3 + 3*e**(log(x)**2)*x* 
*4 - 4*e**(log(x)**2) - x**5 + 4*x),x) - log(x) + x))/2