Integrand size = 123, antiderivative size = 25 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=\frac {x}{-2-x+\log (x) \left (x+\frac {6 \log (x)}{4+8 x}\right )} \] Output:
x/(ln(x)*(x+ln(x)/(4/3*x+2/3))-2-x)
Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=\frac {2 \left (x+2 x^2\right )}{-4-10 x-4 x^2+2 x \log (x)+4 x^2 \log (x)+3 \log ^2(x)} \] Input:
Integrate[(-8 - 36*x - 48*x^2 - 16*x^3 + (-12 - 24*x)*Log[x] + (6 + 24*x)* Log[x]^2)/(16 + 80*x + 132*x^2 + 80*x^3 + 16*x^4 + (-16*x - 72*x^2 - 96*x^ 3 - 32*x^4)*Log[x] + (-24 - 60*x - 20*x^2 + 16*x^3 + 16*x^4)*Log[x]^2 + (1 2*x + 24*x^2)*Log[x]^3 + 9*Log[x]^4),x]
Output:
(2*(x + 2*x^2))/(-4 - 10*x - 4*x^2 + 2*x*Log[x] + 4*x^2*Log[x] + 3*Log[x]^ 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.
| \(\displaystyle \int \frac {-16 x^3-48 x^2-36 x+(24 x+6) \log ^2(x)+(-24 x-12) \log (x)-8}{16 x^4+80 x^3+132 x^2+\left (24 x^2+12 x\right ) \log ^3(x)+\left (16 x^4+16 x^3-20 x^2-60 x-24\right ) \log ^2(x)+\left (-32 x^4-96 x^3-72 x^2-16 x\right ) \log (x)+80 x+9 \log ^4(x)+16} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-4 (x+2) (2 x+1)^2+6 (4 x+1) \log ^2(x)-12 (2 x+1) \log (x)}{\left (2 \left (2 x^2+5 x+2\right )-3 \log ^2(x)-2 x (2 x+1) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 (4 x+1)}{-4 x^2+4 x^2 \log (x)-10 x+3 \log ^2(x)+2 x \log (x)-4}-\frac {4 (2 x+1) \left (-2 x^2+4 x^2 \log (x)-4 x+x \log (x)+3 \log (x)\right )}{\left (-4 x^2+4 x^2 \log (x)-10 x+3 \log ^2(x)+2 x \log (x)-4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \int \frac {x}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx+40 \int \frac {x^2}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx-12 \int \frac {\log (x)}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx-28 \int \frac {x \log (x)}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx-24 \int \frac {x^2 \log (x)}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx+2 \int \frac {1}{4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4}dx+8 \int \frac {x}{4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4}dx+16 \int \frac {x^3}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx-32 \int \frac {x^3 \log (x)}{\left (4 \log (x) x^2-4 x^2+2 \log (x) x-10 x+3 \log ^2(x)-4\right )^2}dx\) |
Input:
Int[(-8 - 36*x - 48*x^2 - 16*x^3 + (-12 - 24*x)*Log[x] + (6 + 24*x)*Log[x] ^2)/(16 + 80*x + 132*x^2 + 80*x^3 + 16*x^4 + (-16*x - 72*x^2 - 96*x^3 - 32 *x^4)*Log[x] + (-24 - 60*x - 20*x^2 + 16*x^3 + 16*x^4)*Log[x]^2 + (12*x + 24*x^2)*Log[x]^3 + 9*Log[x]^4),x]
Output:
$Aborted
Time = 0.93 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56
| method | result | size |
| risch | \(\frac {2 \left (1+2 x \right ) x}{4 x^{2} \ln \left (x \right )+3 \ln \left (x \right )^{2}+2 x \ln \left (x \right )-4 x^{2}-10 x -4}\) | \(39\) |
| default | \(\frac {4 x^{2}+2 x}{4 x^{2} \ln \left (x \right )+3 \ln \left (x \right )^{2}+2 x \ln \left (x \right )-4 x^{2}-10 x -4}\) | \(40\) |
| norman | \(\frac {4 x^{2}+2 x}{4 x^{2} \ln \left (x \right )+3 \ln \left (x \right )^{2}+2 x \ln \left (x \right )-4 x^{2}-10 x -4}\) | \(41\) |
| parallelrisch | \(\frac {12 x^{2}+6 x}{12 x^{2} \ln \left (x \right )+9 \ln \left (x \right )^{2}+6 x \ln \left (x \right )-12 x^{2}-30 x -12}\) | \(42\) |
Input:
int(((24*x+6)*ln(x)^2+(-24*x-12)*ln(x)-16*x^3-48*x^2-36*x-8)/(9*ln(x)^4+(2 4*x^2+12*x)*ln(x)^3+(16*x^4+16*x^3-20*x^2-60*x-24)*ln(x)^2+(-32*x^4-96*x^3 -72*x^2-16*x)*ln(x)+16*x^4+80*x^3+132*x^2+80*x+16),x,method=_RETURNVERBOSE )
Output:
2*(1+2*x)*x/(4*x^2*ln(x)+3*ln(x)^2+2*x*ln(x)-4*x^2-10*x-4)
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=-\frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} - 2 \, {\left (2 \, x^{2} + x\right )} \log \left (x\right ) - 3 \, \log \left (x\right )^{2} + 10 \, x + 4} \] Input:
integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*lo g(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32 *x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algorith m="fricas")
Output:
-2*(2*x^2 + x)/(4*x^2 - 2*(2*x^2 + x)*log(x) - 3*log(x)^2 + 10*x + 4)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=\frac {4 x^{2} + 2 x}{- 4 x^{2} - 10 x + \left (4 x^{2} + 2 x\right ) \log {\left (x \right )} + 3 \log {\left (x \right )}^{2} - 4} \] Input:
integrate(((24*x+6)*ln(x)**2+(-24*x-12)*ln(x)-16*x**3-48*x**2-36*x-8)/(9*l n(x)**4+(24*x**2+12*x)*ln(x)**3+(16*x**4+16*x**3-20*x**2-60*x-24)*ln(x)**2 +(-32*x**4-96*x**3-72*x**2-16*x)*ln(x)+16*x**4+80*x**3+132*x**2+80*x+16),x )
Output:
(4*x**2 + 2*x)/(-4*x**2 - 10*x + (4*x**2 + 2*x)*log(x) + 3*log(x)**2 - 4)
Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=-\frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} - 2 \, {\left (2 \, x^{2} + x\right )} \log \left (x\right ) - 3 \, \log \left (x\right )^{2} + 10 \, x + 4} \] Input:
integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*lo g(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32 *x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algorith m="maxima")
Output:
-2*(2*x^2 + x)/(4*x^2 - 2*(2*x^2 + x)*log(x) - 3*log(x)^2 + 10*x + 4)
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=\frac {2 \, {\left (2 \, x^{2} + x\right )}}{4 \, x^{2} \log \left (x\right ) - 4 \, x^{2} + 2 \, x \log \left (x\right ) + 3 \, \log \left (x\right )^{2} - 10 \, x - 4} \] Input:
integrate(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*lo g(x)^4+(24*x^2+12*x)*log(x)^3+(16*x^4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32 *x^4-96*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x, algorith m="giac")
Output:
2*(2*x^2 + x)/(4*x^2*log(x) - 4*x^2 + 2*x*log(x) + 3*log(x)^2 - 10*x - 4)
Timed out. \[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx=\int -\frac {36\,x+\ln \left (x\right )\,\left (24\,x+12\right )+48\,x^2+16\,x^3-{\ln \left (x\right )}^2\,\left (24\,x+6\right )+8}{80\,x+{\ln \left (x\right )}^3\,\left (24\,x^2+12\,x\right )-\ln \left (x\right )\,\left (32\,x^4+96\,x^3+72\,x^2+16\,x\right )+9\,{\ln \left (x\right )}^4-{\ln \left (x\right )}^2\,\left (-16\,x^4-16\,x^3+20\,x^2+60\,x+24\right )+132\,x^2+80\,x^3+16\,x^4+16} \,d x \] Input:
int(-(36*x + log(x)*(24*x + 12) + 48*x^2 + 16*x^3 - log(x)^2*(24*x + 6) + 8)/(80*x + log(x)^3*(12*x + 24*x^2) - log(x)*(16*x + 72*x^2 + 96*x^3 + 32* x^4) + 9*log(x)^4 - log(x)^2*(60*x + 20*x^2 - 16*x^3 - 16*x^4 + 24) + 132* x^2 + 80*x^3 + 16*x^4 + 16),x)
Output:
int(-(36*x + log(x)*(24*x + 12) + 48*x^2 + 16*x^3 - log(x)^2*(24*x + 6) + 8)/(80*x + log(x)^3*(12*x + 24*x^2) - log(x)*(16*x + 72*x^2 + 96*x^3 + 32* x^4) + 9*log(x)^4 - log(x)^2*(60*x + 20*x^2 - 16*x^3 - 16*x^4 + 24) + 132* x^2 + 80*x^3 + 16*x^4 + 16), x)
\[ \int \frac {-8-36 x-48 x^2-16 x^3+(-12-24 x) \log (x)+(6+24 x) \log ^2(x)}{16+80 x+132 x^2+80 x^3+16 x^4+\left (-16 x-72 x^2-96 x^3-32 x^4\right ) \log (x)+\left (-24-60 x-20 x^2+16 x^3+16 x^4\right ) \log ^2(x)+\left (12 x+24 x^2\right ) \log ^3(x)+9 \log ^4(x)} \, dx =\text {Too large to display} \] Input:
int(((24*x+6)*log(x)^2+(-24*x-12)*log(x)-16*x^3-48*x^2-36*x-8)/(9*log(x)^4 +(24*x^2+12*x)*log(x)^3+(16*x^4+16*x^3-20*x^2-60*x-24)*log(x)^2+(-32*x^4-9 6*x^3-72*x^2-16*x)*log(x)+16*x^4+80*x^3+132*x^2+80*x+16),x)
Output:
2*(3*int(log(x)**2/(9*log(x)**4 + 24*log(x)**3*x**2 + 12*log(x)**3*x + 16* log(x)**2*x**4 + 16*log(x)**2*x**3 - 20*log(x)**2*x**2 - 60*log(x)**2*x - 24*log(x)**2 - 32*log(x)*x**4 - 96*log(x)*x**3 - 72*log(x)*x**2 - 16*log(x )*x + 16*x**4 + 80*x**3 + 132*x**2 + 80*x + 16),x) - 8*int(x**3/(9*log(x)* *4 + 24*log(x)**3*x**2 + 12*log(x)**3*x + 16*log(x)**2*x**4 + 16*log(x)**2 *x**3 - 20*log(x)**2*x**2 - 60*log(x)**2*x - 24*log(x)**2 - 32*log(x)*x**4 - 96*log(x)*x**3 - 72*log(x)*x**2 - 16*log(x)*x + 16*x**4 + 80*x**3 + 132 *x**2 + 80*x + 16),x) - 24*int(x**2/(9*log(x)**4 + 24*log(x)**3*x**2 + 12* log(x)**3*x + 16*log(x)**2*x**4 + 16*log(x)**2*x**3 - 20*log(x)**2*x**2 - 60*log(x)**2*x - 24*log(x)**2 - 32*log(x)*x**4 - 96*log(x)*x**3 - 72*log(x )*x**2 - 16*log(x)*x + 16*x**4 + 80*x**3 + 132*x**2 + 80*x + 16),x) - 6*in t(log(x)/(9*log(x)**4 + 24*log(x)**3*x**2 + 12*log(x)**3*x + 16*log(x)**2* x**4 + 16*log(x)**2*x**3 - 20*log(x)**2*x**2 - 60*log(x)**2*x - 24*log(x)* *2 - 32*log(x)*x**4 - 96*log(x)*x**3 - 72*log(x)*x**2 - 16*log(x)*x + 16*x **4 + 80*x**3 + 132*x**2 + 80*x + 16),x) + 12*int((log(x)**2*x)/(9*log(x)* *4 + 24*log(x)**3*x**2 + 12*log(x)**3*x + 16*log(x)**2*x**4 + 16*log(x)**2 *x**3 - 20*log(x)**2*x**2 - 60*log(x)**2*x - 24*log(x)**2 - 32*log(x)*x**4 - 96*log(x)*x**3 - 72*log(x)*x**2 - 16*log(x)*x + 16*x**4 + 80*x**3 + 132 *x**2 + 80*x + 16),x) - 12*int((log(x)*x)/(9*log(x)**4 + 24*log(x)**3*x**2 + 12*log(x)**3*x + 16*log(x)**2*x**4 + 16*log(x)**2*x**3 - 20*log(x)**...