Integrand size = 177, antiderivative size = 29 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=-16+x+\frac {\log (x)}{x-\frac {x}{\left (x+4 x^2\right )^2-\log (x)}} \] Output:
ln(x)/(x-x/((4*x^2+x)^2-ln(x)))-16+x
Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {-1+x^2+\frac {-1+x^2+8 x^3+16 x^4}{-1+x^2+8 x^3+16 x^4-\log (x)}+\log (x)}{x} \] Input:
Integrate[(-8*x^3 - 17*x^4 + 65*x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^ 10 + (2 - x^2 - 32*x^3 - 83*x^4 - 32*x^5 - 128*x^6 - 256*x^7 - 256*x^8)*Lo g[x] + (3*x^2 + 16*x^3 + 32*x^4)*Log[x]^2 - Log[x]^3)/(x^2 - 2*x^4 - 16*x^ 5 - 31*x^6 + 16*x^7 + 96*x^8 + 256*x^9 + 256*x^10 + (2*x^2 - 2*x^4 - 16*x^ 5 - 32*x^6)*Log[x] + x^2*Log[x]^2),x]
Output:
(-1 + x^2 + (-1 + x^2 + 8*x^3 + 16*x^4)/(-1 + x^2 + 8*x^3 + 16*x^4 - Log[x ]) + 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.
| \(\displaystyle \int \frac {256 x^{10}+256 x^9+352 x^8+272 x^7+65 x^6-17 x^4-8 x^3+\left (32 x^4+16 x^3+3 x^2\right ) \log ^2(x)+\left (-256 x^8-256 x^7-128 x^6-32 x^5-83 x^4-32 x^3-x^2+2\right ) \log (x)-\log ^3(x)}{256 x^{10}+256 x^9+96 x^8+16 x^7-31 x^6-16 x^5-2 x^4+x^2+x^2 \log ^2(x)+\left (-32 x^6-16 x^5-2 x^4+2 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {256 x^{10}+256 x^9+352 x^8+272 x^7+65 x^6-17 x^4-8 x^3+\left (32 x^4+16 x^3+3 x^2\right ) \log ^2(x)+\left (-256 x^8-256 x^7-128 x^6-32 x^5-83 x^4-32 x^3-x^2+2\right ) \log (x)-\log ^3(x)}{x^2 \left (-16 x^4-8 x^3-x^2+\log (x)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2+2}{x^2}-\frac {\log (x)}{x^2}+\frac {48 x^4+16 x^3+x^2+1}{x^2 \left (16 x^4+8 x^3+x^2-\log (x)-1\right )}+\frac {-1024 x^8-896 x^7-288 x^6-40 x^5+78 x^4+32 x^3+3 x^2-1}{x^2 \left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx-\int \frac {1}{x^2 \left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx+32 \int \frac {x}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx+78 \int \frac {x^2}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx-40 \int \frac {x^3}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx-288 \int \frac {x^4}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx+\int \frac {1}{16 x^4+8 x^3+x^2-\log (x)-1}dx+\int \frac {1}{x^2 \left (16 x^4+8 x^3+x^2-\log (x)-1\right )}dx+16 \int \frac {x}{16 x^4+8 x^3+x^2-\log (x)-1}dx+48 \int \frac {x^2}{16 x^4+8 x^3+x^2-\log (x)-1}dx-1024 \int \frac {x^6}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx-896 \int \frac {x^5}{\left (16 x^4+8 x^3+x^2-\log (x)-1\right )^2}dx+x-\frac {1}{x}+\frac {\log (x)}{x}\) |
Input:
Int[(-8*x^3 - 17*x^4 + 65*x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^10 + ( 2 - x^2 - 32*x^3 - 83*x^4 - 32*x^5 - 128*x^6 - 256*x^7 - 256*x^8)*Log[x] + (3*x^2 + 16*x^3 + 32*x^4)*Log[x]^2 - Log[x]^3)/(x^2 - 2*x^4 - 16*x^5 - 31 *x^6 + 16*x^7 + 96*x^8 + 256*x^9 + 256*x^10 + (2*x^2 - 2*x^4 - 16*x^5 - 32 *x^6)*Log[x] + x^2*Log[x]^2),x]
Output:
$Aborted
Time = 7.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97
| method | result | size |
| risch | \(\frac {\ln \left (x \right )}{x}+\frac {x^{2}-1}{x}+\frac {16 x^{4}+8 x^{3}+x^{2}-1}{x \left (16 x^{4}+8 x^{3}+x^{2}-\ln \left (x \right )-1\right )}\) | \(57\) |
| default | \(x +\frac {\ln \left (x \right )^{2}}{\left (\ln \left (x \right )+1\right ) x}-\frac {\ln \left (x \right ) x \left (16 x^{2}+8 x +1\right )}{\left (\ln \left (x \right )+1\right ) \left (-16 x^{4}-8 x^{3}-x^{2}+\ln \left (x \right )+1\right )}\) | \(59\) |
| parallelrisch | \(\frac {16 x^{4} \ln \left (x \right )+8 x^{3} \ln \left (x \right )-\ln \left (x \right )^{2}+x^{4}-x^{2}+16 x^{6}+8 x^{5}}{x \left (16 x^{4}+8 x^{3}+x^{2}-\ln \left (x \right )-1\right )}\) | \(65\) |
Input:
int((-ln(x)^3+(32*x^4+16*x^3+3*x^2)*ln(x)^2+(-256*x^8-256*x^7-128*x^6-32*x ^5-83*x^4-32*x^3-x^2+2)*ln(x)+256*x^10+256*x^9+352*x^8+272*x^7+65*x^6-17*x ^4-8*x^3)/(x^2*ln(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*ln(x)+256*x^10+256*x^9 +96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x,method=_RETURNVERBOSE)
Output:
ln(x)/x+(x^2-1)/x+(16*x^4+8*x^3+x^2-1)/x/(16*x^4+8*x^3+x^2-ln(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {16 \, x^{6} + 8 \, x^{5} + x^{4} - x^{2} + 8 \, {\left (2 \, x^{4} + x^{3}\right )} \log \left (x\right ) - \log \left (x\right )^{2}}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \left (x\right ) - x} \] Input:
integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128* x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(x)+256*x^10+256*x^9+352*x^8+272*x^7+65 *x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256*x ^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="fricas")
Output:
(16*x^6 + 8*x^5 + x^4 - x^2 + 8*(2*x^4 + x^3)*log(x) - log(x)^2)/(16*x^5 + 8*x^4 + x^3 - x*log(x) - x)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=x + \frac {- 16 x^{4} - 8 x^{3} - x^{2} + 1}{- 16 x^{5} - 8 x^{4} - x^{3} + x \log {\left (x \right )} + x} + \frac {\log {\left (x \right )}}{x} - \frac {1}{x} \] Input:
integrate((-ln(x)**3+(32*x**4+16*x**3+3*x**2)*ln(x)**2+(-256*x**8-256*x**7 -128*x**6-32*x**5-83*x**4-32*x**3-x**2+2)*ln(x)+256*x**10+256*x**9+352*x** 8+272*x**7+65*x**6-17*x**4-8*x**3)/(x**2*ln(x)**2+(-32*x**6-16*x**5-2*x**4 +2*x**2)*ln(x)+256*x**10+256*x**9+96*x**8+16*x**7-31*x**6-16*x**5-2*x**4+x **2),x)
Output:
x + (-16*x**4 - 8*x**3 - x**2 + 1)/(-16*x**5 - 8*x**4 - x**3 + x*log(x) + x) + log(x)/x - 1/x
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=\frac {16 \, x^{6} + 8 \, x^{5} + x^{4} - x^{2} + 8 \, {\left (2 \, x^{4} + x^{3}\right )} \log \left (x\right ) - \log \left (x\right )^{2}}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \left (x\right ) - x} \] Input:
integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128* x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(x)+256*x^10+256*x^9+352*x^8+272*x^7+65 *x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256*x ^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="maxima")
Output:
(16*x^6 + 8*x^5 + x^4 - x^2 + 8*(2*x^4 + x^3)*log(x) - log(x)^2)/(16*x^5 + 8*x^4 + x^3 - x*log(x) - x)
Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=x + \frac {16 \, x^{4} + 8 \, x^{3} + x^{2} - 1}{16 \, x^{5} + 8 \, x^{4} + x^{3} - x \log \left (x\right ) - x} + \frac {\log \left (x\right )}{x} - \frac {1}{x} \] Input:
integrate((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128* x^6-32*x^5-83*x^4-32*x^3-x^2+2)*log(x)+256*x^10+256*x^9+352*x^8+272*x^7+65 *x^6-17*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256*x ^10+256*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x, algorithm="giac")
Output:
x + (16*x^4 + 8*x^3 + x^2 - 1)/(16*x^5 + 8*x^4 + x^3 - x*log(x) - x) + log (x)/x - 1/x
Time = 1.74 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.83 \[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=x+\frac {\ln \left (x\right )}{x}-\frac {\frac {x^4}{4}+\frac {x^3}{8}+\frac {x^2}{64}-\frac {1}{32}}{x^5+\frac {3\,x^4}{8}+\frac {x^3}{32}-\frac {x}{64}}+\frac {\frac {256\,x^8+256\,x^7+96\,x^6+16\,x^5-47\,x^4-24\,x^3-3\,x^2+2}{x\,\left (64\,x^4+24\,x^3+2\,x^2-1\right )}+\frac {\ln \left (x\right )\,\left (48\,x^4+16\,x^3+x^2+1\right )}{x\,\left (64\,x^4+24\,x^3+2\,x^2-1\right )}}{x^2-\ln \left (x\right )+8\,x^3+16\,x^4-1} \] Input:
int((log(x)^2*(3*x^2 + 16*x^3 + 32*x^4) - log(x)^3 - log(x)*(x^2 + 32*x^3 + 83*x^4 + 32*x^5 + 128*x^6 + 256*x^7 + 256*x^8 - 2) - 8*x^3 - 17*x^4 + 65 *x^6 + 272*x^7 + 352*x^8 + 256*x^9 + 256*x^10)/(x^2*log(x)^2 - log(x)*(2*x ^4 - 2*x^2 + 16*x^5 + 32*x^6) + x^2 - 2*x^4 - 16*x^5 - 31*x^6 + 16*x^7 + 9 6*x^8 + 256*x^9 + 256*x^10),x)
Output:
x + log(x)/x - (x^2/64 + x^3/8 + x^4/4 - 1/32)/(x^3/32 - x/64 + (3*x^4)/8 + x^5) + ((16*x^5 - 24*x^3 - 47*x^4 - 3*x^2 + 96*x^6 + 256*x^7 + 256*x^8 + 2)/(x*(2*x^2 + 24*x^3 + 64*x^4 - 1)) + (log(x)*(x^2 + 16*x^3 + 48*x^4 + 1 ))/(x*(2*x^2 + 24*x^3 + 64*x^4 - 1)))/(x^2 - log(x) + 8*x^3 + 16*x^4 - 1)
\[ \int \frac {-8 x^3-17 x^4+65 x^6+272 x^7+352 x^8+256 x^9+256 x^{10}+\left (2-x^2-32 x^3-83 x^4-32 x^5-128 x^6-256 x^7-256 x^8\right ) \log (x)+\left (3 x^2+16 x^3+32 x^4\right ) \log ^2(x)-\log ^3(x)}{x^2-2 x^4-16 x^5-31 x^6+16 x^7+96 x^8+256 x^9+256 x^{10}+\left (2 x^2-2 x^4-16 x^5-32 x^6\right ) \log (x)+x^2 \log ^2(x)} \, dx=\text {too large to display} \] Input:
int((-log(x)^3+(32*x^4+16*x^3+3*x^2)*log(x)^2+(-256*x^8-256*x^7-128*x^6-32 *x^5-83*x^4-32*x^3-x^2+2)*log(x)+256*x^10+256*x^9+352*x^8+272*x^7+65*x^6-1 7*x^4-8*x^3)/(x^2*log(x)^2+(-32*x^6-16*x^5-2*x^4+2*x^2)*log(x)+256*x^10+25 6*x^9+96*x^8+16*x^7-31*x^6-16*x^5-2*x^4+x^2),x)
Output:
- int(log(x)**3/(log(x)**2*x**2 - 32*log(x)*x**6 - 16*log(x)*x**5 - 2*log (x)*x**4 + 2*log(x)*x**2 + 256*x**10 + 256*x**9 + 96*x**8 + 16*x**7 - 31*x **6 - 16*x**5 - 2*x**4 + x**2),x) + 3*int(log(x)**2/(log(x)**2 - 32*log(x) *x**4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2*log(x) + 256*x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) + 256*int(x**8/(log (x)**2 - 32*log(x)*x**4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2*log(x) + 256* x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) + 256*int(x**7/(log(x)**2 - 32*log(x)*x**4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2*log(x) + 256*x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) + 352*int(x**6/(log(x)**2 - 32*log(x)*x**4 - 16*log(x)*x* *3 - 2*log(x)*x**2 + 2*log(x) + 256*x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) + 272*int(x**5/(log(x)**2 - 32*log(x)*x **4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2*log(x) + 256*x**8 + 256*x**7 + 96 *x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) + 65*int(x**4/(log(x) **2 - 32*log(x)*x**4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2*log(x) + 256*x** 8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2*x**2 + 1),x) - 17 *int(x**2/(log(x)**2 - 32*log(x)*x**4 - 16*log(x)*x**3 - 2*log(x)*x**2 + 2 *log(x) + 256*x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*x**4 - 16*x**3 - 2* x**2 + 1),x) - int(log(x)/(log(x)**2 - 32*log(x)*x**4 - 16*log(x)*x**3 - 2 *log(x)*x**2 + 2*log(x) + 256*x**8 + 256*x**7 + 96*x**6 + 16*x**5 - 31*...