Integrand size = 120, antiderivative size = 20 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3 x^2 \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right ) \]
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3 x^2 \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right ) \]
Integrate[(9*x + 12*x^2 + 18*x^3 - 6*x^4 - 6*x^2*Log[x] + (24*x^2 - 12*x^4 - 12*x^2*Log[x] + (-36*x + 18*x^3 + 18*x*Log[x])*Log[-2 + x^2 + Log[x]])* Log[-2*x + 3*Log[-2 + x^2 + Log[x]]])/(4*x - 2*x^3 - 2*x*Log[x] + (-6 + 3* x^2 + 3*Log[x])*Log[-2 + x^2 + Log[x]]),x]
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.
\(\displaystyle \int \frac {-6 x^4+18 x^3+12 x^2-6 x^2 \log (x)+\left (-12 x^4+24 x^2-12 x^2 \log (x)+\left (18 x^3-36 x+18 x \log (x)\right ) \log \left (x^2+\log (x)-2\right )\right ) \log \left (3 \log \left (x^2+\log (x)-2\right )-2 x\right )+9 x}{-2 x^3+\left (3 x^2+3 \log (x)-6\right ) \log \left (x^2+\log (x)-2\right )+4 x-2 x \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-6 x^4+18 x^3+12 x^2-6 x^2 \log (x)+\left (-12 x^4+24 x^2-12 x^2 \log (x)+\left (18 x^3-36 x+18 x \log (x)\right ) \log \left (x^2+\log (x)-2\right )\right ) \log \left (3 \log \left (x^2+\log (x)-2\right )-2 x\right )+9 x}{\left (-x^2-\log (x)+2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 x^2 \log (x)}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}-\frac {12 x^2}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}+6 x \log \left (3 \log \left (x^2+\log (x)-2\right )-2 x\right )-\frac {9 x}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}+\frac {6 x^4}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}-\frac {18 x^3}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -9 \int \frac {x}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx-12 \int \frac {x^2}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx+6 \int \frac {x^2 \log (x)}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx+6 \int x \log \left (3 \log \left (x^2+\log (x)-2\right )-2 x\right )dx+6 \int \frac {x^4}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx-18 \int \frac {x^3}{\left (x^2+\log (x)-2\right ) \left (2 x-3 \log \left (x^2+\log (x)-2\right )\right )}dx\) |
Int[(9*x + 12*x^2 + 18*x^3 - 6*x^4 - 6*x^2*Log[x] + (24*x^2 - 12*x^4 - 12* x^2*Log[x] + (-36*x + 18*x^3 + 18*x*Log[x])*Log[-2 + x^2 + Log[x]])*Log[-2 *x + 3*Log[-2 + x^2 + Log[x]]])/(4*x - 2*x^3 - 2*x*Log[x] + (-6 + 3*x^2 + 3*Log[x])*Log[-2 + x^2 + Log[x]]),x]
3.13.65.3.1 Defintions of rubi rules used
Time = 19.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05
method | result | size |
risch | \(3 x^{2} \ln \left (3 \ln \left (\ln \left (x \right )+x^{2}-2\right )-2 x \right )\) | \(21\) |
parallelrisch | \(3 x^{2} \ln \left (3 \ln \left (\ln \left (x \right )+x^{2}-2\right )-2 x \right )\) | \(21\) |
int((((18*x*ln(x)+18*x^3-36*x)*ln(ln(x)+x^2-2)-12*x^2*ln(x)-12*x^4+24*x^2) *ln(3*ln(ln(x)+x^2-2)-2*x)-6*x^2*ln(x)-6*x^4+18*x^3+12*x^2+9*x)/((3*ln(x)+ 3*x^2-6)*ln(ln(x)+x^2-2)-2*x*ln(x)-2*x^3+4*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \left (x\right ) - 2\right )\right ) \]
integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x ^4+24*x^2)*log(3*log(log(x)+x^2-2)-2*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9 *x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algorit hm=\
Exception generated. \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=\text {Exception raised: TypeError} \]
integrate((((18*x*ln(x)+18*x**3-36*x)*ln(ln(x)+x**2-2)-12*x**2*ln(x)-12*x* *4+24*x**2)*ln(3*ln(ln(x)+x**2-2)-2*x)-6*x**2*ln(x)-6*x**4+18*x**3+12*x**2 +9*x)/((3*ln(x)+3*x**2-6)*ln(ln(x)+x**2-2)-2*x*ln(x)-2*x**3+4*x),x)
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \left (x\right ) - 2\right )\right ) \]
integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x ^4+24*x^2)*log(3*log(log(x)+x^2-2)-2*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9 *x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algorit hm=\
Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3 \, x^{2} \log \left (-2 \, x + 3 \, \log \left (x^{2} + \log \left (x\right ) - 2\right )\right ) \]
integrate((((18*x*log(x)+18*x^3-36*x)*log(log(x)+x^2-2)-12*x^2*log(x)-12*x ^4+24*x^2)*log(3*log(log(x)+x^2-2)-2*x)-6*x^2*log(x)-6*x^4+18*x^3+12*x^2+9 *x)/((3*log(x)+3*x^2-6)*log(log(x)+x^2-2)-2*x*log(x)-2*x^3+4*x),x, algorit hm=\
Time = 11.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+12 x^2+18 x^3-6 x^4-6 x^2 \log (x)+\left (24 x^2-12 x^4-12 x^2 \log (x)+\left (-36 x+18 x^3+18 x \log (x)\right ) \log \left (-2+x^2+\log (x)\right )\right ) \log \left (-2 x+3 \log \left (-2+x^2+\log (x)\right )\right )}{4 x-2 x^3-2 x \log (x)+\left (-6+3 x^2+3 \log (x)\right ) \log \left (-2+x^2+\log (x)\right )} \, dx=3\,x^2\,\ln \left (3\,\ln \left (\ln \left (x\right )+x^2-2\right )-2\,x\right ) \]