Integrand size = 193, antiderivative size = 27 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=-2+x+\left (-x+\frac {2}{x \left (\frac {1}{2}-x^2+\log (x)\right )}\right )^2 \] Output:
(2/x/(ln(x)+1/2-x^2)-x)^2-2+x
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=x+x^2+\frac {16}{x^2 \left (1-2 x^2+2 \log (x)\right )^2}-\frac {8}{1-2 x^2+2 \log (x)} \] Input:
Integrate[(-96 + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + (-64 + 32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24 *x^7 + 48*x^8)*Log[x] + (12*x^3 + 24*x^4 - 24*x^5 - 48*x^6)*Log[x]^2 + (8* x^3 + 16*x^4)*Log[x]^3)/(x^3 - 6*x^5 + 12*x^7 - 8*x^9 + (6*x^3 - 24*x^5 + 24*x^7)*Log[x] + (12*x^3 - 24*x^5)*Log[x]^2 + 8*x^3*Log[x]^3),x]
Output:
x + x^2 + 16/(x^2*(1 - 2*x^2 + 2*Log[x])^2) - 8/(1 - 2*x^2 + 2*Log[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.
| \(\displaystyle \int \frac {-16 x^{10}-8 x^9+24 x^8+12 x^7+52 x^6-6 x^5-62 x^4+x^3+208 x^2+\left (16 x^4+8 x^3\right ) \log ^3(x)+\left (-48 x^6-24 x^5+24 x^4+12 x^3\right ) \log ^2(x)+\left (48 x^8+24 x^7-48 x^6-24 x^5-52 x^4+6 x^3+32 x^2-64\right ) \log (x)-96}{-8 x^9+12 x^7-6 x^5+x^3+8 x^3 \log ^3(x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+\left (24 x^7-24 x^5+6 x^3\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-16 x^{10}-8 x^9+24 x^8+12 x^7+52 x^6-6 x^5-62 x^4+x^3+208 x^2+\left (16 x^4+8 x^3\right ) \log ^3(x)+\left (-48 x^6-24 x^5+24 x^4+12 x^3\right ) \log ^2(x)+\left (48 x^8+24 x^7-48 x^6-24 x^5-52 x^4+6 x^3+32 x^2-64\right ) \log (x)-96}{x^3 \left (-2 x^2+2 \log (x)+1\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {64 \left (2 x^2-1\right )}{x^3 \left (2 x^2-2 \log (x)-1\right )^3}-\frac {16 \left (2 x^4-x^2+2\right )}{x^3 \left (2 x^2-2 \log (x)-1\right )^2}+2 x+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -128 \int \frac {1}{x \left (2 x^2-2 \log (x)-1\right )^3}dx+16 \int \frac {1}{x \left (2 x^2-2 \log (x)-1\right )^2}dx-32 \int \frac {x}{\left (2 x^2-2 \log (x)-1\right )^2}dx+64 \int \frac {1}{x^3 \left (2 x^2-2 \log (x)-1\right )^3}dx-32 \int \frac {1}{x^3 \left (2 x^2-2 \log (x)-1\right )^2}dx+x^2+x\) |
Input:
Int[(-96 + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x ^9 - 16*x^10 + (-64 + 32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8)*Log[x] + (12*x^3 + 24*x^4 - 24*x^5 - 48*x^6)*Log[x]^2 + (8*x^3 + 16*x^4)*Log[x]^3)/(x^3 - 6*x^5 + 12*x^7 - 8*x^9 + (6*x^3 - 24*x^5 + 24*x^7 )*Log[x] + (12*x^3 - 24*x^5)*Log[x]^2 + 8*x^3*Log[x]^3),x]
Output:
$Aborted
Time = 15.71 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(x^{2}+x +\frac {16 x^{4}-16 x^{2} \ln \left (x \right )-8 x^{2}+16}{x^{2} \left (2 x^{2}-2 \ln \left (x \right )-1\right )^{2}}\) | \(43\) |
| default | \(x^{2}+x +\frac {16}{\left (2 \ln \left (x \right )+1\right )^{2} x^{2}}-\frac {8 \left (-8 x^{2} \ln \left (x \right )^{2}+8 \ln \left (x \right )^{3}-8 x^{2} \ln \left (x \right )+12 \ln \left (x \right )^{2}+6 x^{2}-10 \ln \left (x \right )-7\right )}{\left (2 \ln \left (x \right )+1\right )^{2} \left (-2 x^{2}+2 \ln \left (x \right )+1\right )^{2}}\) | \(81\) |
| parallelrisch | \(\frac {64+16 x^{4} \ln \left (x \right )^{2}-32 x^{5} \ln \left (x \right )-32 x^{6} \ln \left (x \right )+16 x^{4} \ln \left (x \right )+16 x^{3} \ln \left (x \right )^{2}+16 x^{8}+16 x^{7}+16 x^{3} \ln \left (x \right )-64 x^{2} \ln \left (x \right )-16 x^{6}-32 x^{2}+4 x^{3}+68 x^{4}-16 x^{5}}{4 x^{2} \left (4 x^{4}-8 x^{2} \ln \left (x \right )-4 x^{2}+4 \ln \left (x \right )^{2}+4 \ln \left (x \right )+1\right )}\) | \(127\) |
Input:
int(((16*x^4+8*x^3)*ln(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*ln(x)^2+(48*x^8 +24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+32*x^2-64)*ln(x)-16*x^10-8*x^9+24*x^8+1 2*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*ln(x)^3+(-24*x^5+12*x^3)* ln(x)^2+(24*x^7-24*x^5+6*x^3)*ln(x)-8*x^9+12*x^7-6*x^5+x^3),x,method=_RETU RNVERBOSE)
Output:
x^2+x+8*(2*x^4-2*x^2*ln(x)-x^2+2)/x^2/(2*x^2-2*ln(x)-1)^2
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=\frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \left (x\right )^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \left (x\right )} \] Input:
integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2 +(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9 +24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*x^ 5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="fricas")
Output:
(4*x^8 + 4*x^7 - 4*x^6 - 4*x^5 + 17*x^4 + x^3 + 4*(x^4 + x^3)*log(x)^2 - 8 *x^2 - 4*(2*x^6 + 2*x^5 - x^4 - x^3 + 4*x^2)*log(x) + 16)/(4*x^6 - 4*x^4 + 4*x^2*log(x)^2 + x^2 - 4*(2*x^4 - x^2)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=x^{2} + x + \frac {16 x^{4} - 16 x^{2} \log {\left (x \right )} - 8 x^{2} + 16}{4 x^{6} - 4 x^{4} + 4 x^{2} \log {\left (x \right )}^{2} + x^{2} + \left (- 8 x^{4} + 4 x^{2}\right ) \log {\left (x \right )}} \] Input:
integrate(((16*x**4+8*x**3)*ln(x)**3+(-48*x**6-24*x**5+24*x**4+12*x**3)*ln (x)**2+(48*x**8+24*x**7-48*x**6-24*x**5-52*x**4+6*x**3+32*x**2-64)*ln(x)-1 6*x**10-8*x**9+24*x**8+12*x**7+52*x**6-6*x**5-62*x**4+x**3+208*x**2-96)/(8 *x**3*ln(x)**3+(-24*x**5+12*x**3)*ln(x)**2+(24*x**7-24*x**5+6*x**3)*ln(x)- 8*x**9+12*x**7-6*x**5+x**3),x)
Output:
x**2 + x + (16*x**4 - 16*x**2*log(x) - 8*x**2 + 16)/(4*x**6 - 4*x**4 + 4*x **2*log(x)**2 + x**2 + (-8*x**4 + 4*x**2)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=\frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \left (x\right )^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \left (x\right )} \] Input:
integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2 +(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9 +24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*x^ 5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="maxima")
Output:
(4*x^8 + 4*x^7 - 4*x^6 - 4*x^5 + 17*x^4 + x^3 + 4*(x^4 + x^3)*log(x)^2 - 8 *x^2 - 4*(2*x^6 + 2*x^5 - x^4 - x^3 + 4*x^2)*log(x) + 16)/(4*x^6 - 4*x^4 + 4*x^2*log(x)^2 + x^2 - 4*(2*x^4 - x^2)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (25) = 50\).
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=x^{2} + x + \frac {8 \, {\left (2 \, x^{4} - 2 \, x^{2} \log \left (x\right ) - x^{2} + 2\right )}}{4 \, x^{6} - 8 \, x^{4} \log \left (x\right ) - 4 \, x^{4} + 4 \, x^{2} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (x\right ) + x^{2}} \] Input:
integrate(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2 +(48*x^8+24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9 +24*x^8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*x^ 5+12*x^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x, algorithm="giac")
Output:
x^2 + x + 8*(2*x^4 - 2*x^2*log(x) - x^2 + 2)/(4*x^6 - 8*x^4*log(x) - 4*x^4 + 4*x^2*log(x)^2 + 4*x^2*log(x) + x^2)
Timed out. \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (48\,x^8+24\,x^7-48\,x^6-24\,x^5-52\,x^4+6\,x^3+32\,x^2-64\right )+{\ln \left (x\right )}^3\,\left (16\,x^4+8\,x^3\right )+208\,x^2+x^3-62\,x^4-6\,x^5+52\,x^6+12\,x^7+24\,x^8-8\,x^9-16\,x^{10}+{\ln \left (x\right )}^2\,\left (-48\,x^6-24\,x^5+24\,x^4+12\,x^3\right )-96}{\ln \left (x\right )\,\left (24\,x^7-24\,x^5+6\,x^3\right )+{\ln \left (x\right )}^2\,\left (12\,x^3-24\,x^5\right )+8\,x^3\,{\ln \left (x\right )}^3+x^3-6\,x^5+12\,x^7-8\,x^9} \,d x \] Input:
int((log(x)*(32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8 - 64) + log(x)^3*(8*x^3 + 16*x^4) + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + log(x)^2*(12*x^3 + 24*x^4 - 24*x^5 - 48*x^6) - 96)/(log(x)*(6*x^3 - 24*x^5 + 24*x^7) + log(x)^2*(12*x^3 - 24* x^5) + 8*x^3*log(x)^3 + x^3 - 6*x^5 + 12*x^7 - 8*x^9),x)
Output:
int((log(x)*(32*x^2 + 6*x^3 - 52*x^4 - 24*x^5 - 48*x^6 + 24*x^7 + 48*x^8 - 64) + log(x)^3*(8*x^3 + 16*x^4) + 208*x^2 + x^3 - 62*x^4 - 6*x^5 + 52*x^6 + 12*x^7 + 24*x^8 - 8*x^9 - 16*x^10 + log(x)^2*(12*x^3 + 24*x^4 - 24*x^5 - 48*x^6) - 96)/(log(x)*(6*x^3 - 24*x^5 + 24*x^7) + log(x)^2*(12*x^3 - 24* x^5) + 8*x^3*log(x)^3 + x^3 - 6*x^5 + 12*x^7 - 8*x^9), x)
Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.81 \[ \int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx=\frac {16 \mathrm {log}\left (x \right )^{2} x^{4}+16 \mathrm {log}\left (x \right )^{2} x^{3}+68 \mathrm {log}\left (x \right )^{2} x^{2}-32 \,\mathrm {log}\left (x \right ) x^{6}-32 \,\mathrm {log}\left (x \right ) x^{5}-120 \,\mathrm {log}\left (x \right ) x^{4}+16 \,\mathrm {log}\left (x \right ) x^{3}+4 \,\mathrm {log}\left (x \right ) x^{2}+16 x^{8}+16 x^{7}+52 x^{6}-16 x^{5}+4 x^{3}-15 x^{2}+64}{4 x^{2} \left (4 \mathrm {log}\left (x \right )^{2}-8 \,\mathrm {log}\left (x \right ) x^{2}+4 \,\mathrm {log}\left (x \right )+4 x^{4}-4 x^{2}+1\right )} \] Input:
int(((16*x^4+8*x^3)*log(x)^3+(-48*x^6-24*x^5+24*x^4+12*x^3)*log(x)^2+(48*x ^8+24*x^7-48*x^6-24*x^5-52*x^4+6*x^3+32*x^2-64)*log(x)-16*x^10-8*x^9+24*x^ 8+12*x^7+52*x^6-6*x^5-62*x^4+x^3+208*x^2-96)/(8*x^3*log(x)^3+(-24*x^5+12*x ^3)*log(x)^2+(24*x^7-24*x^5+6*x^3)*log(x)-8*x^9+12*x^7-6*x^5+x^3),x)
Output:
(16*log(x)**2*x**4 + 16*log(x)**2*x**3 + 68*log(x)**2*x**2 - 32*log(x)*x** 6 - 32*log(x)*x**5 - 120*log(x)*x**4 + 16*log(x)*x**3 + 4*log(x)*x**2 + 16 *x**8 + 16*x**7 + 52*x**6 - 16*x**5 + 4*x**3 - 15*x**2 + 64)/(4*x**2*(4*lo g(x)**2 - 8*log(x)*x**2 + 4*log(x) + 4*x**4 - 4*x**2 + 1))