Integrand size = 139, antiderivative size = 22 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^2}{\log \left ((4+x)^2 \left (-\frac {1}{x^2}+x\right )^2\right )} \] Output:
x^2/ln((4+x)^2*(x-1/x^2)^2)
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^2}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \] Input:
Integrate[(-16*x - 2*x^2 - 8*x^4 - 4*x^5 + (-8*x - 2*x^2 + 8*x^4 + 2*x^5)* Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8)/x^4] )/((-4 - x + 4*x^3 + x^4)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8)/x^4]^2),x]
Output:
x^2/Log[(-4 - x + 4*x^3 + x^4)^2/x^4]
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.
\(\displaystyle \int \frac {-4 x^5-8 x^4-2 x^2+\left (2 x^5+8 x^4-2 x^2-8 x\right ) \log \left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )-16 x}{\left (x^4+4 x^3-x-4\right ) \log ^2\left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-4 x^5-8 x^4-2 x^2+\left (2 x^5+8 x^4-2 x^2-8 x\right ) \log \left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )-16 x}{15 (x-1) \log ^2\left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )}-\frac {-4 x^5-8 x^4-2 x^2+\left (2 x^5+8 x^4-2 x^2-8 x\right ) \log \left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )-16 x}{65 (x+4) \log ^2\left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )}+\frac {(-2 x-7) \left (-4 x^5-8 x^4-2 x^2+\left (2 x^5+8 x^4-2 x^2-8 x\right ) \log \left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )-16 x\right )}{39 \left (x^2+x+1\right ) \log ^2\left (\frac {x^8+8 x^7+16 x^6-2 x^5-16 x^4-32 x^3+x^2+8 x+16}{x^4}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \int \frac {1}{\log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx-\frac {4 i \int \frac {1}{\left (-2 x+i \sqrt {3}-1\right ) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx}{\sqrt {3}}-2 \int \frac {1}{(x-1) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx-4 \int \frac {x}{\log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx-32 \int \frac {1}{(x+4) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx+\frac {2}{3} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}+1\right ) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx+\frac {2}{3} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx-\frac {4 i \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \log ^2\left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx}{\sqrt {3}}+2 \int \frac {x}{\log \left (\frac {\left (x^4+4 x^3-x-4\right )^2}{x^4}\right )}dx\) |
Input:
Int[(-16*x - 2*x^2 - 8*x^4 - 4*x^5 + (-8*x - 2*x^2 + 8*x^4 + 2*x^5)*Log[(1 6 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8)/x^4])/((-4 - x + 4*x^3 + x^4)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8)/x^4]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 2.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18
method | result | size |
norman | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
risch | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
parallelrisch | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
Input:
int(((2*x^5+8*x^4-2*x^2-8*x)*ln((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+ 8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)/(x^4+4*x^3-x-4)/ln((x^8+8*x^7+16*x^6- 2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x,method=_RETURNVERBOSE)
Output:
x^2/ln((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \] Input:
integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x ^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7 +16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="fricas")
Output:
x^2/log((x^8 + 8*x^7 + 16*x^6 - 2*x^5 - 16*x^4 - 32*x^3 + x^2 + 8*x + 16)/ x^4)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {x^{8} + 8 x^{7} + 16 x^{6} - 2 x^{5} - 16 x^{4} - 32 x^{3} + x^{2} + 8 x + 16}{x^{4}} \right )}} \] Input:
integrate(((2*x**5+8*x**4-2*x**2-8*x)*ln((x**8+8*x**7+16*x**6-2*x**5-16*x* *4-32*x**3+x**2+8*x+16)/x**4)-4*x**5-8*x**4-2*x**2-16*x)/(x**4+4*x**3-x-4) /ln((x**8+8*x**7+16*x**6-2*x**5-16*x**4-32*x**3+x**2+8*x+16)/x**4)**2,x)
Output:
x**2/log((x**8 + 8*x**7 + 16*x**6 - 2*x**5 - 16*x**4 - 32*x**3 + x**2 + 8* x + 16)/x**4)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{2 \, {\left (\log \left (x^{2} + x + 1\right ) + \log \left (x + 4\right ) + \log \left (x - 1\right ) - 2 \, \log \left (x\right )\right )}} \] Input:
integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x ^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7 +16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="maxima")
Output:
1/2*x^2/(log(x^2 + x + 1) + log(x + 4) + log(x - 1) - 2*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \] Input:
integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x ^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7 +16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="giac")
Output:
x^2/log((x^8 + 8*x^7 + 16*x^6 - 2*x^5 - 16*x^4 - 32*x^3 + x^2 + 8*x + 16)/ x^4)
Time = 3.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 7.27 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=x+\frac {x^2+\frac {x^2\,\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )\,\left (-x^4-4\,x^3+x+4\right )}{2\,x^4+4\,x^3+x+8}}{\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )}-\frac {-\frac {13\,x^3}{4}+\frac {9\,x^2}{2}+3\,x-8}{x^4+2\,x^3+\frac {x}{2}+4}+\frac {x^2}{2} \] Input:
int((16*x + log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^ 8 + 16)/x^4)*(8*x + 2*x^2 - 8*x^4 - 2*x^5) + 2*x^2 + 8*x^4 + 4*x^5)/(log(( 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4)^2*(x - 4*x^3 - x^4 + 4)),x)
Output:
x + (x^2 + (x^2*log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4)*(x - 4*x^3 - x^4 + 4))/(x + 4*x^3 + 2*x^4 + 8))/log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4) - (3*x + (9*x^2)/2 - (13*x^3)/4 - 8)/(x/2 + 2*x^3 + x^4 + 4) + x^2/2
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\mathrm {log}\left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )} \] Input:
int(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2 +8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7+16*x^ 6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x)
Output:
x**2/log((x**8 + 8*x**7 + 16*x**6 - 2*x**5 - 16*x**4 - 32*x**3 + x**2 + 8* x + 16)/x**4)