Integrand size = 63, antiderivative size = 18 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\log \left (x+\frac {x}{\frac {16}{x^4}+x}\right )+\log (\log (x)) \] Output:
ln(x/(x+16/x^4)+x)+ln(ln(x))
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\log (x)-\log \left (16+x^5\right )+\log \left (16+x^4+x^5\right )+\log (\log (x)) \] Input:
Integrate[(256 + 16*x^4 + 32*x^5 + x^9 + x^10 + (256 + 80*x^4 + 32*x^5 + x ^10)*Log[x])/((256*x + 16*x^5 + 32*x^6 + x^10 + x^11)*Log[x]),x]
Output:
Log[x] - Log[16 + x^5] + Log[16 + x^4 + x^5] + Log[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.
\(\displaystyle \int \frac {x^{10}+x^9+32 x^5+16 x^4+\left (x^{10}+32 x^5+80 x^4+256\right ) \log (x)+256}{\left (x^{11}+x^{10}+32 x^6+16 x^5+256 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^{10}+x^9+32 x^5+16 x^4+\left (x^{10}+32 x^5+80 x^4+256\right ) \log (x)+256}{x \left (x^{10}+x^9+32 x^5+16 x^4+256\right ) \log (x)}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {x^{10}+x^9+32 x^5+16 x^4+\left (x^{10}+32 x^5+80 x^4+256\right ) \log (x)+256}{768 x (x+2) \log (x)}-\frac {x^{10}+x^9+32 x^5+16 x^4+\left (x^{10}+32 x^5+80 x^4+256\right ) \log (x)+256}{16 \left (x^5+16\right ) \log (x)}+\frac {\left (x^3-3 x^2+8 x+28\right ) \left (x^{10}+x^9+32 x^5+16 x^4+\left (x^{10}+32 x^5+80 x^4+256\right ) \log (x)+256\right )}{768 x \left (x^4-x^3+2 x^2-4 x+8\right ) \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{768} \int \frac {x^9-x^8+2 x^7-4 x^6+8 x^5+16 x^4-16 x^3+32 x^2-64 x+128}{x \log (x)}dx+\frac {1}{768} \int \frac {x^9-x^8+2 x^7+44 x^6+56 x^5+16 x^4-16 x^3+32 x^2+704 x+896}{x \log (x)}dx-\frac {1}{16} \operatorname {ExpIntegralEi}(5 \log (x))-\frac {1}{16} \operatorname {ExpIntegralEi}(6 \log (x))-\operatorname {LogIntegral}(x)-\log \left (x^5+16\right )+\log \left (x^4-x^3+2 x^2-4 x+8\right )+\log (x)+\log (x+2)\) |
Input:
Int[(256 + 16*x^4 + 32*x^5 + x^9 + x^10 + (256 + 80*x^4 + 32*x^5 + x^10)*L og[x])/((256*x + 16*x^5 + 32*x^6 + x^10 + x^11)*Log[x]),x]
Output:
$Aborted
Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33
method | result | size |
risch | \(-\ln \left (x^{5}+16\right )+\ln \left (x^{6}+x^{5}+16 x \right )+\ln \left (\ln \left (x \right )\right )\) | \(24\) |
default | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \right )+\ln \left (2+x \right )-\ln \left (x^{5}+16\right )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )\) | \(38\) |
norman | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \right )+\ln \left (2+x \right )-\ln \left (x^{5}+16\right )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )\) | \(38\) |
parallelrisch | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \right )+\ln \left (2+x \right )-\ln \left (x^{5}+16\right )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )\) | \(38\) |
parts | \(\ln \left (\ln \left (x \right )\right )+\ln \left (x \right )+\ln \left (2+x \right )-\ln \left (x^{5}+16\right )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )\) | \(38\) |
Input:
int(((x^10+32*x^5+80*x^4+256)*ln(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^10 +32*x^6+16*x^5+256*x)/ln(x),x,method=_RETURNVERBOSE)
Output:
-ln(x^5+16)+ln(x^6+x^5+16*x)+ln(ln(x))
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\log \left (x^{6} + x^{5} + 16 \, x\right ) - \log \left (x^{5} + 16\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^ 11+x^10+32*x^6+16*x^5+256*x)/log(x),x, algorithm="fricas")
Output:
log(x^6 + x^5 + 16*x) - log(x^5 + 16) + log(log(x))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=- \log {\left (x^{5} + 16 \right )} + \log {\left (x^{6} + x^{5} + 16 x \right )} + \log {\left (\log {\left (x \right )} \right )} \] Input:
integrate(((x**10+32*x**5+80*x**4+256)*ln(x)+x**10+x**9+32*x**5+16*x**4+25 6)/(x**11+x**10+32*x**6+16*x**5+256*x)/ln(x),x)
Output:
-log(x**5 + 16) + log(x**6 + x**5 + 16*x) + log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=-\log \left (x^{5} + 16\right ) + \log \left (x^{4} - x^{3} + 2 \, x^{2} - 4 \, x + 8\right ) + \log \left (x + 2\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^ 11+x^10+32*x^6+16*x^5+256*x)/log(x),x, algorithm="maxima")
Output:
-log(x^5 + 16) + log(x^4 - x^3 + 2*x^2 - 4*x + 8) + log(x + 2) + log(x) + log(log(x))
Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\log \left (x^{5} + x^{4} + 16\right ) - \log \left (x^{5} + 16\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^ 11+x^10+32*x^6+16*x^5+256*x)/log(x),x, algorithm="giac")
Output:
log(x^5 + x^4 + 16) - log(x^5 + 16) + log(x) + log(log(x))
Time = 3.90 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\ln \left (\ln \left (x\right )\right )+\ln \left (x\,\left (x^5+x^4+16\right )\right )-\ln \left (x^5+16\right ) \] Input:
int((16*x^4 + 32*x^5 + x^9 + x^10 + log(x)*(80*x^4 + 32*x^5 + x^10 + 256) + 256)/(log(x)*(256*x + 16*x^5 + 32*x^6 + x^10 + x^11)),x)
Output:
log(log(x)) + log(x*(x^4 + x^5 + 16)) - log(x^5 + 16)
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.44 \[ \int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{\left (256 x+16 x^5+32 x^6+x^{10}+x^{11}\right ) \log (x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (x^{9}-x^{8}+2 x^{7}-4 x^{6}+8 x^{5}+16 x^{4}-16 x^{3}+32 x^{2}-64 x +128\right )-2 \,\mathrm {log}\left (x^{5}+16\right )+\mathrm {log}\left (x +2\right )+\mathrm {log}\left (x \right ) \] Input:
int(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^1 0+32*x^6+16*x^5+256*x)/log(x),x)
Output:
log(log(x)) + log(x**9 - x**8 + 2*x**7 - 4*x**6 + 8*x**5 + 16*x**4 - 16*x* *3 + 32*x**2 - 64*x + 128) - 2*log(x**5 + 16) + log(x + 2) + log(x)