Integrand size = 141, antiderivative size = 30 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \left (x-\log \left (x \left (-1+\frac {4}{x^2 \left (-e^{\log ^2(x)}+x\right )^2}\right )\right )\right ) \]
Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \left (x+2 \log \left (e^{\log ^2(x)}-x\right )+\log (x)-\log \left (2+e^{\log ^2(x)} x-x^2\right )-\log \left (2-e^{\log ^2(x)} x+x^2\right )\right ) \]
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]
(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
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 |
| \(\displaystyle -\frac {3}{2} \left (2 \int \frac {1}{x \left (x^2-e^{\log ^2(x)} x-2\right )}dx+\int \frac {x}{x^2-e^{\log ^2(x)} x-2}dx-2 \int \frac {1}{x \left (x^2-e^{\log ^2(x)} x+2\right )}dx+\int \frac {x}{x^2-e^{\log ^2(x)} x+2}dx+4 \int \frac {\log (x)}{x \left (x^2-e^{\log ^2(x)} x-2\right )}dx-2 \int \frac {x \log (x)}{x^2-e^{\log ^2(x)} x-2}dx-4 \int \frac {\log (x)}{x \left (x^2-e^{\log ^2(x)} x+2\right )}dx-2 \int \frac {x \log (x)}{x^2-e^{\log ^2(x)} x+2}dx+2 \int \frac {1}{e^{\log ^2(x)}-x}dx-4 \int \frac {\log (x)}{e^{\log ^2(x)}-x}dx-x+\log (x)\right )\) |
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]
3.10.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.30 (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 | \(\frac {3 x}{2}+\frac {3 \ln \left (x \right )}{2}+3 \ln \left (x -{\mathrm e}^{\ln \left (x \right )^{2}}\right )-\frac {3 \ln \left (x^{2}-{\mathrm e}^{\ln \left (x \right )^{2}} x -2\right )}{2}-\frac {3 \ln \left (x^{2}-{\mathrm e}^{\ln \left (x \right )^{2}} x +2\right )}{2}\) | \(53\) |
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)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x\right ) + 3 \, \log \left (-x + e^{\left (\log \left (x\right )^{2}\right )}\right ) - \frac {3}{2} \, \log \left (\frac {x^{4} - 2 \, x^{3} e^{\left (\log \left (x\right )^{2}\right )} + x^{2} e^{\left (2 \, \log \left (x\right )^{2}\right )} - 4}{x^{2}}\right ) \]
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=\
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)
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3 x}{2} - \frac {3 \log {\left (x \right )}}{2} + 3 \log {\left (- x + e^{\log {\left (x \right )}^{2}} \right )} - \frac {3 \log {\left (- 2 x e^{\log {\left (x \right )}^{2}} + e^{2 \log {\left (x \right )}^{2}} + \frac {x^{4} - 4}{x^{2}} \right )}}{2} \]
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)
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
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x\right ) + 3 \, \log \left (-x + e^{\left (\log \left (x\right )^{2}\right )}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \left (x\right )^{2}\right )} + 2}{x}\right ) - \frac {3}{2} \, \log \left (-\frac {x^{2} - x e^{\left (\log \left (x\right )^{2}\right )} - 2}{x}\right ) \]
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=\
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)
Time = 4.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\frac {3}{2} \, x - \frac {3}{2} \, \log \left (x^{4} - 2 \, x^{3} e^{\left (\log \left (x\right )^{2}\right )} + x^{2} e^{\left (2 \, \log \left (x\right )^{2}\right )} - 4\right ) + 3 \, \log \left (x - e^{\left (\log \left (x\right )^{2}\right )}\right ) + \frac {3}{2} \, \log \left (x\right ) \]
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=\
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)
Timed out. \[ \int \frac {36 x+12 x^2+3 x^5-3 x^6+e^{3 \log ^2(x)} \left (-3 x^2+3 x^3\right )+e^{2 \log ^2(x)} \left (9 x^3-9 x^4\right )+e^{\log ^2(x)} \left (-12-12 x-9 x^4+9 x^5-48 \log (x)\right )}{8 x^2+2 e^{3 \log ^2(x)} x^3-6 e^{2 \log ^2(x)} x^4-2 x^6+e^{\log ^2(x)} \left (-8 x+6 x^5\right )} \, dx=\int -\frac {36\,x-{\mathrm {e}}^{3\,{\ln \left (x\right )}^2}\,\left (3\,x^2-3\,x^3\right )+{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,\left (9\,x^3-9\,x^4\right )+12\,x^2+3\,x^5-3\,x^6-{\mathrm {e}}^{{\ln \left (x\right )}^2}\,\left (12\,x+48\,\ln \left (x\right )+9\,x^4-9\,x^5+12\right )}{6\,x^4\,{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}-2\,x^3\,{\mathrm {e}}^{3\,{\ln \left (x\right )}^2}+{\mathrm {e}}^{{\ln \left (x\right )}^2}\,\left (8\,x-6\,x^5\right )-8\,x^2+2\,x^6} \,d x \]
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)