Integrand size = 124, antiderivative size = 23 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x+\left (-5+2 x+x^2+\log ^2(x)+\log ^2\left (\log ^4(x)\right )\right )^2 \] Output:
(2*x+ln(x)^2-5+x^2+ln(ln(x)^4)^2)^2+x
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(23)=46\).
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=-19 x-6 x^2+4 x^3+x^4+2 \left (-5+2 x+x^2\right ) \log ^2(x)+\log ^4(x)+2 \left (-5+2 x+x^2+\log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right ) \] Input:
Integrate[((-19*x - 12*x^2 + 12*x^3 + 4*x^4)*Log[x] + (-20 + 8*x + 4*x^2)* Log[x]^2 + (4*x + 4*x^2)*Log[x]^3 + 4*Log[x]^4 + (-80 + 32*x + 16*x^2 + 16 *Log[x]^2)*Log[Log[x]^4] + ((4*x + 4*x^2)*Log[x] + 4*Log[x]^2)*Log[Log[x]^ 4]^2 + 16*Log[Log[x]^4]^3)/(x*Log[x]),x]
Output:
-19*x - 6*x^2 + 4*x^3 + x^4 + 2*(-5 + 2*x + x^2)*Log[x]^2 + Log[x]^4 + 2*( -5 + 2*x + x^2 + Log[x]^2)*Log[Log[x]^4]^2 + Log[Log[x]^4]^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.
\(\displaystyle \int \frac {\left (4 x^2+4 x\right ) \log ^3(x)+\left (4 x^2+8 x-20\right ) \log ^2(x)+\left (\left (4 x^2+4 x\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+\left (16 x^2+32 x+16 \log ^2(x)-80\right ) \log \left (\log ^4(x)\right )+\left (4 x^4+12 x^3-12 x^2-19 x\right ) \log (x)+4 \log ^4(x)+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \left (x^2+x+\log (x)\right ) \log ^2\left (\log ^4(x)\right )}{x}+\frac {16 \left (x^2+2 x+\log ^2(x)-5\right ) \log \left (\log ^4(x)\right )}{x \log (x)}+\frac {4 x^4+12 x^3-12 x^2+4 x^2 \log ^2(x)+4 x^2 \log (x)-19 x+4 \log ^3(x)+4 x \log ^2(x)+8 x \log (x)-20 \log (x)}{x}+\frac {16 \log ^3\left (\log ^4(x)\right )}{x \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)}dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)}dx+4 \int \log ^2\left (\log ^4(x)\right )dx+4 \int x \log ^2\left (\log ^4(x)\right )dx+x^4+4 x^3-6 x^2+2 x^2 \log ^2(x)-19 x+\log ^4(x)+\log ^4\left (\log ^4(x)\right )+4 x \log ^2(x)-10 \log ^2(x)+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )-10 \log ^2\left (\log ^4(x)\right )\) |
Input:
Int[((-19*x - 12*x^2 + 12*x^3 + 4*x^4)*Log[x] + (-20 + 8*x + 4*x^2)*Log[x] ^2 + (4*x + 4*x^2)*Log[x]^3 + 4*Log[x]^4 + (-80 + 32*x + 16*x^2 + 16*Log[x ]^2)*Log[Log[x]^4] + ((4*x + 4*x^2)*Log[x] + 4*Log[x]^2)*Log[Log[x]^4]^2 + 16*Log[Log[x]^4]^3)/(x*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(23)=46\).
Time = 1.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.13
method | result | size |
parallelrisch | \(x^{4}+2 x^{2} \ln \left (x \right )^{2}+2 \ln \left (\ln \left (x \right )^{4}\right )^{2} x^{2}+\ln \left (x \right )^{4}+2 \ln \left (x \right )^{2} \ln \left (\ln \left (x \right )^{4}\right )^{2}+\ln \left (\ln \left (x \right )^{4}\right )^{4}+4 x^{3}+4 x \ln \left (x \right )^{2}+4 x \ln \left (\ln \left (x \right )^{4}\right )^{2}-6 x^{2}-10 \ln \left (x \right )^{2}-10 \ln \left (\ln \left (x \right )^{4}\right )^{2}-19 x\) | \(95\) |
risch | \(\text {Expression too large to display}\) | \(12852\) |
Input:
int((16*ln(ln(x)^4)^3+(4*ln(x)^2+(4*x^2+4*x)*ln(x))*ln(ln(x)^4)^2+(16*ln(x )^2+16*x^2+32*x-80)*ln(ln(x)^4)+4*ln(x)^4+(4*x^2+4*x)*ln(x)^3+(4*x^2+8*x-2 0)*ln(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*ln(x))/x/ln(x),x,method=_RETURNVERBO SE)
Output:
x^4+2*x^2*ln(x)^2+2*ln(ln(x)^4)^2*x^2+ln(x)^4+2*ln(x)^2*ln(ln(x)^4)^2+ln(l n(x)^4)^4+4*x^3+4*x*ln(x)^2+4*x*ln(ln(x)^4)^2-6*x^2-10*ln(x)^2-10*ln(ln(x) ^4)^2-19*x
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (\log \left (x\right )^{4}\right )^{4} + \log \left (x\right )^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \left (x\right )^{2} - 6 \, x^{2} - 19 \, x \] Input:
integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4 )^2+(16*log(x)^2+16*x^2+32*x-80)*log(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log( x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x),x , algorithm="fricas")
Output:
x^4 + log(log(x)^4)^4 + log(x)^4 + 4*x^3 + 2*(x^2 + log(x)^2 + 2*x - 5)*lo g(log(x)^4)^2 + 2*(x^2 + 2*x - 5)*log(x)^2 - 6*x^2 - 19*x
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + 4 x^{3} - 6 x^{2} - 19 x + \left (2 x^{2} + 4 x - 10\right ) \log {\left (x \right )}^{2} + \left (2 x^{2} + 4 x + 2 \log {\left (x \right )}^{2} - 10\right ) \log {\left (\log {\left (x \right )}^{4} \right )}^{2} + \log {\left (x \right )}^{4} + \log {\left (\log {\left (x \right )}^{4} \right )}^{4} \] Input:
integrate((16*ln(ln(x)**4)**3+(4*ln(x)**2+(4*x**2+4*x)*ln(x))*ln(ln(x)**4) **2+(16*ln(x)**2+16*x**2+32*x-80)*ln(ln(x)**4)+4*ln(x)**4+(4*x**2+4*x)*ln( x)**3+(4*x**2+8*x-20)*ln(x)**2+(4*x**4+12*x**3-12*x**2-19*x)*ln(x))/x/ln(x ),x)
Output:
x**4 + 4*x**3 - 6*x**2 - 19*x + (2*x**2 + 4*x - 10)*log(x)**2 + (2*x**2 + 4*x + 2*log(x)**2 - 10)*log(log(x)**4)**2 + log(x)**4 + log(log(x)**4)**4
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.09 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (x\right )^{4} + 256 \, \log \left (\log \left (x\right )\right )^{4} + {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + 4 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 32 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - 7 \, x^{2} + 8 \, x \log \left (x\right ) - 10 \, \log \left (x\right )^{2} - 27 \, x \] Input:
integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4 )^2+(16*log(x)^2+16*x^2+32*x-80)*log(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log( x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x),x , algorithm="maxima")
Output:
x^4 + log(x)^4 + 256*log(log(x))^4 + (2*log(x)^2 - 2*log(x) + 1)*x^2 + 4*x ^3 + 2*x^2*log(x) + 32*(x^2 + log(x)^2 + 2*x - 5)*log(log(x))^2 + 4*(log(x )^2 - 2*log(x) + 2)*x - 7*x^2 + 8*x*log(x) - 10*log(x)^2 - 27*x
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (\log \left (x\right )^{4}\right )^{4} + \log \left (x\right )^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \left (x\right )^{2} - 6 \, x^{2} - 19 \, x \] Input:
integrate((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4 )^2+(16*log(x)^2+16*x^2+32*x-80)*log(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log( x)^3+(4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x),x , algorithm="giac")
Output:
x^4 + log(log(x)^4)^4 + log(x)^4 + 4*x^3 + 2*(x^2 + log(x)^2 + 2*x - 5)*lo g(log(x)^4)^2 + 2*(x^2 + 2*x - 5)*log(x)^2 - 6*x^2 - 19*x
Time = 3.82 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx={\ln \left ({\ln \left (x\right )}^4\right )}^4-19\,x+{\ln \left (x\right )}^2\,\left (2\,x^2+4\,x-10\right )+{\ln \left (x\right )}^4+{\ln \left ({\ln \left (x\right )}^4\right )}^2\,\left (2\,x^2+4\,x+2\,{\ln \left (x\right )}^2-10\right )-6\,x^2+4\,x^3+x^4 \] Input:
int((16*log(log(x)^4)^3 + log(x)^3*(4*x + 4*x^2) + log(x)^2*(8*x + 4*x^2 - 20) - log(x)*(19*x + 12*x^2 - 12*x^3 - 4*x^4) + log(log(x)^4)*(32*x + 16* log(x)^2 + 16*x^2 - 80) + log(log(x)^4)^2*(4*log(x)^2 + log(x)*(4*x + 4*x^ 2)) + 4*log(x)^4)/(x*log(x)),x)
Output:
log(log(x)^4)^4 - 19*x + log(x)^2*(4*x + 2*x^2 - 10) + log(x)^4 + log(log( x)^4)^2*(4*x + 2*log(x)^2 + 2*x^2 - 10) - 6*x^2 + 4*x^3 + x^4
Time = 0.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.09 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )^{4}\right )^{4}+2 \mathrm {log}\left (\mathrm {log}\left (x \right )^{4}\right )^{2} \mathrm {log}\left (x \right )^{2}+2 \mathrm {log}\left (\mathrm {log}\left (x \right )^{4}\right )^{2} x^{2}+4 \mathrm {log}\left (\mathrm {log}\left (x \right )^{4}\right )^{2} x -10 \mathrm {log}\left (\mathrm {log}\left (x \right )^{4}\right )^{2}+\mathrm {log}\left (x \right )^{4}+2 \mathrm {log}\left (x \right )^{2} x^{2}+4 \mathrm {log}\left (x \right )^{2} x -10 \mathrm {log}\left (x \right )^{2}+x^{4}+4 x^{3}-6 x^{2}-19 x \] Input:
int((16*log(log(x)^4)^3+(4*log(x)^2+(4*x^2+4*x)*log(x))*log(log(x)^4)^2+(1 6*log(x)^2+16*x^2+32*x-80)*log(log(x)^4)+4*log(x)^4+(4*x^2+4*x)*log(x)^3+( 4*x^2+8*x-20)*log(x)^2+(4*x^4+12*x^3-12*x^2-19*x)*log(x))/x/log(x),x)
Output:
log(log(x)**4)**4 + 2*log(log(x)**4)**2*log(x)**2 + 2*log(log(x)**4)**2*x* *2 + 4*log(log(x)**4)**2*x - 10*log(log(x)**4)**2 + log(x)**4 + 2*log(x)** 2*x**2 + 4*log(x)**2*x - 10*log(x)**2 + x**4 + 4*x**3 - 6*x**2 - 19*x