Integrand size = 133, antiderivative size = 25 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {1}{6} \log ^2(5) \left (\log (x)+4 x \left (1+\frac {x}{\log (\log (x))}\right )\right )^2 \] Output:
1/6*ln(5)^2*(ln(x)+4*(x/ln(ln(x))+1)*x)^2
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {\log ^2(5) \left (4 x^2+(4 x+\log (x)) \log (\log (x))\right )^2}{6 \log ^2(\log (x))} \] Input:
Integrate[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5] ^2*Log[x])*Log[Log[x]] + ((4*x^2 + 48*x^3)*Log[5]^2*Log[x] + 8*x^2*Log[5]^ 2*Log[x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*Lo g[5]^2*Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]
Output:
(Log[5]^2*(4*x^2 + (4*x + Log[x])*Log[Log[x]])^2)/(6*Log[Log[x]]^2)
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.
| \(\displaystyle \int \frac {-16 x^4 \log ^2(5)+\left (\left (16 x^2+4 x\right ) \log ^2(5) \log (x)+(4 x+1) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))+\left (8 x^2 \log ^2(5) \log ^2(x)+\left (48 x^3+4 x^2\right ) \log ^2(5) \log (x)\right ) \log ^2(\log (x))+\left (\left (32 x^4-4 x^2\right ) \log ^2(5) \log (x)-16 x^3 \log ^2(5)\right ) \log (\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {16 \log ^2(5) x^4-\left ((4 x+1) \log ^2(5) \log ^2(x)+4 \left (4 x^2+x\right ) \log ^2(5) \log (x)\right ) \log ^3(\log (x))-4 \left (2 x^2 \log ^2(5) \log ^2(x)+\left (12 x^3+x^2\right ) \log ^2(5) \log (x)\right ) \log ^2(\log (x))+4 \left (4 \log ^2(5) x^3+\left (x^2-8 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))}{x \log (x) \log ^3(\log (x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {16 \log ^2(5) x^4-\left ((4 x+1) \log ^2(5) \log ^2(x)+4 \left (4 x^2+x\right ) \log ^2(5) \log (x)\right ) \log ^3(\log (x))-4 \left (2 x^2 \log ^2(5) \log ^2(x)+\left (12 x^3+x^2\right ) \log ^2(5) \log (x)\right ) \log ^2(\log (x))+4 \left (4 \log ^2(5) x^3+\left (x^2-8 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))}{x \log (x) \log ^3(\log (x))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (\frac {16 \log ^2(5) x^3}{\log (x) \log ^3(\log (x))}-\frac {4 \log ^2(5) (12 x+2 \log (x)+1) x}{\log (\log (x))}-\frac {4 \log ^2(5) \left (8 \log (x) x^2-4 x-\log (x)\right ) x}{\log (x) \log ^2(\log (x))}-\frac {(4 x+1) \log ^2(5) (4 x+\log (x))}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (32 \log ^2(5) \int \frac {x^3}{\log ^2(\log (x))}dx-16 \log ^2(5) \int \frac {x^3}{\log (x) \log ^3(\log (x))}dx-16 \log ^2(5) \int \frac {x^2}{\log (x) \log ^2(\log (x))}dx+48 \log ^2(5) \int \frac {x^2}{\log (\log (x))}dx-4 \log ^2(5) \int \frac {x}{\log ^2(\log (x))}dx+4 \log ^2(5) \int \frac {x}{\log (\log (x))}dx+8 \log ^2(5) \int \frac {x \log (x)}{\log (\log (x))}dx+\frac {1}{2} \log ^2(5) (4 x+\log (x))^2\right )\) |
Input:
Int[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5]^2*Log [x])*Log[Log[x]] + ((4*x^2 + 48*x^3)*Log[5]^2*Log[x] + 8*x^2*Log[5]^2*Log[ x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*Log[5]^2 *Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(23)=46\).
Time = 1.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48
| method | result | size |
| risch | \(\frac {\ln \left (5\right )^{2} \ln \left (x \right )^{2}}{6}+\frac {4 \ln \left (x \right ) \ln \left (5\right )^{2} x}{3}+\frac {8 x^{2} \ln \left (5\right )^{2}}{3}+\frac {4 x^{2} \ln \left (5\right )^{2} \left (2 x^{2}+4 x \ln \left (\ln \left (x \right )\right )+\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )\right )}{3 \ln \left (\ln \left (x \right )\right )^{2}}\) | \(62\) |
| parallelrisch | \(\frac {16 x^{4} \ln \left (5\right )^{2}+32 \ln \left (5\right )^{2} x^{3} \ln \left (\ln \left (x \right )\right )+8 \ln \left (5\right )^{2} x^{2} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+16 \ln \left (5\right )^{2} \ln \left (\ln \left (x \right )\right )^{2} x^{2}+8 x \ln \left (5\right )^{2} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}+\ln \left (5\right )^{2} \ln \left (\ln \left (x \right )\right )^{2} \ln \left (x \right )^{2}}{6 \ln \left (\ln \left (x \right )\right )^{2}}\) | \(86\) |
Input:
int(1/3*(((1+4*x)*ln(5)^2*ln(x)^2+(16*x^2+4*x)*ln(5)^2*ln(x))*ln(ln(x))^3+ (8*x^2*ln(5)^2*ln(x)^2+(48*x^3+4*x^2)*ln(5)^2*ln(x))*ln(ln(x))^2+((32*x^4- 4*x^2)*ln(5)^2*ln(x)-16*x^3*ln(5)^2)*ln(ln(x))-16*x^4*ln(5)^2)/x/ln(x)/ln( ln(x))^3,x,method=_RETURNVERBOSE)
Output:
1/6*ln(5)^2*ln(x)^2+4/3*ln(x)*ln(5)^2*x+8/3*x^2*ln(5)^2+4/3*x^2*ln(5)^2*(2 *x^2+4*x*ln(ln(x))+ln(x)*ln(ln(x)))/ln(ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (23) = 46\).
Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {16 \, x^{4} \log \left (5\right )^{2} + {\left (16 \, x^{2} \log \left (5\right )^{2} + 8 \, x \log \left (5\right )^{2} \log \left (x\right ) + \log \left (5\right )^{2} \log \left (x\right )^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + 8 \, {\left (4 \, x^{3} \log \left (5\right )^{2} + x^{2} \log \left (5\right )^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )}{6 \, \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(1/3*(((1+4*x)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*lo g(log(x))^3+(8*x^2*log(5)^2*log(x)^2+(48*x^3+4*x^2)*log(5)^2*log(x))*log(l og(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16*x ^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="fricas")
Output:
1/6*(16*x^4*log(5)^2 + (16*x^2*log(5)^2 + 8*x*log(5)^2*log(x) + log(5)^2*l og(x)^2)*log(log(x))^2 + 8*(4*x^3*log(5)^2 + x^2*log(5)^2*log(x))*log(log( x)))/log(log(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {8 x^{2} \log {\left (5 \right )}^{2}}{3} + \frac {4 x \log {\left (5 \right )}^{2} \log {\left (x \right )}}{3} + \frac {8 x^{4} \log {\left (5 \right )}^{2} + \left (16 x^{3} \log {\left (5 \right )}^{2} + 4 x^{2} \log {\left (5 \right )}^{2} \log {\left (x \right )}\right ) \log {\left (\log {\left (x \right )} \right )}}{3 \log {\left (\log {\left (x \right )} \right )}^{2}} + \frac {\log {\left (5 \right )}^{2} \log {\left (x \right )}^{2}}{6} \] Input:
integrate(1/3*(((1+4*x)*ln(5)**2*ln(x)**2+(16*x**2+4*x)*ln(5)**2*ln(x))*ln (ln(x))**3+(8*x**2*ln(5)**2*ln(x)**2+(48*x**3+4*x**2)*ln(5)**2*ln(x))*ln(l n(x))**2+((32*x**4-4*x**2)*ln(5)**2*ln(x)-16*x**3*ln(5)**2)*ln(ln(x))-16*x **4*ln(5)**2)/x/ln(x)/ln(ln(x))**3,x)
Output:
8*x**2*log(5)**2/3 + 4*x*log(5)**2*log(x)/3 + (8*x**4*log(5)**2 + (16*x**3 *log(5)**2 + 4*x**2*log(5)**2*log(x))*log(log(x)))/(3*log(log(x))**2) + lo g(5)**2*log(x)**2/6
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.28 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {8}{3} \, x^{2} \log \left (5\right )^{2} + \frac {1}{6} \, \log \left (5\right )^{2} \log \left (x\right )^{2} + \frac {4}{3} \, {\left (x \log \left (x\right ) - x\right )} \log \left (5\right )^{2} + \frac {4}{3} \, x \log \left (5\right )^{2} + \frac {4 \, {\left (2 \, x^{4} \log \left (5\right )^{2} + {\left (4 \, x^{3} \log \left (5\right )^{2} + x^{2} \log \left (5\right )^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )\right )}}{3 \, \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(1/3*(((1+4*x)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*lo g(log(x))^3+(8*x^2*log(5)^2*log(x)^2+(48*x^3+4*x^2)*log(5)^2*log(x))*log(l og(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16*x ^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="maxima")
Output:
8/3*x^2*log(5)^2 + 1/6*log(5)^2*log(x)^2 + 4/3*(x*log(x) - x)*log(5)^2 + 4 /3*x*log(5)^2 + 4/3*(2*x^4*log(5)^2 + (4*x^3*log(5)^2 + x^2*log(5)^2*log(x ))*log(log(x)))/log(log(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (23) = 46\).
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {8}{3} \, x^{2} \log \left (5\right )^{2} + \frac {4}{3} \, x \log \left (5\right )^{2} \log \left (x\right ) + \frac {1}{6} \, \log \left (5\right )^{2} \log \left (x\right )^{2} + \frac {4 \, {\left (2 \, x^{4} \log \left (5\right )^{2} + 4 \, x^{3} \log \left (5\right )^{2} \log \left (\log \left (x\right )\right ) + x^{2} \log \left (5\right )^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )\right )}}{3 \, \log \left (\log \left (x\right )\right )^{2}} \] Input:
integrate(1/3*(((1+4*x)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*lo g(log(x))^3+(8*x^2*log(5)^2*log(x)^2+(48*x^3+4*x^2)*log(5)^2*log(x))*log(l og(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16*x ^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="giac")
Output:
8/3*x^2*log(5)^2 + 4/3*x*log(5)^2*log(x) + 1/6*log(5)^2*log(x)^2 + 4/3*(2* x^4*log(5)^2 + 4*x^3*log(5)^2*log(log(x)) + x^2*log(5)^2*log(x)*log(log(x) ))/log(log(x))^2
Time = 3.56 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.92 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {8\,x^2\,{\ln \left (5\right )}^2}{3}+\frac {{\ln \left (5\right )}^2\,{\ln \left (x\right )}^2}{6}+\frac {4\,x\,{\ln \left (5\right )}^2\,\ln \left (x\right )}{3}+\frac {16\,x^3\,{\ln \left (5\right )}^2}{3\,\ln \left (\ln \left (x\right )\right )}+\frac {8\,x^4\,{\ln \left (5\right )}^2}{3\,{\ln \left (\ln \left (x\right )\right )}^2}+\frac {4\,x^2\,{\ln \left (5\right )}^2\,\ln \left (x\right )}{3\,\ln \left (\ln \left (x\right )\right )} \] Input:
int(-((16*x^4*log(5)^2)/3 - (log(log(x))^3*(log(5)^2*log(x)*(4*x + 16*x^2) + log(5)^2*log(x)^2*(4*x + 1)))/3 - (log(log(x))^2*(8*x^2*log(5)^2*log(x) ^2 + log(5)^2*log(x)*(4*x^2 + 48*x^3)))/3 + (log(log(x))*(16*x^3*log(5)^2 + log(5)^2*log(x)*(4*x^2 - 32*x^4)))/3)/(x*log(log(x))^3*log(x)),x)
Output:
(8*x^2*log(5)^2)/3 + (log(5)^2*log(x)^2)/6 + (4*x*log(5)^2*log(x))/3 + (16 *x^3*log(5)^2)/(3*log(log(x))) + (8*x^4*log(5)^2)/(3*log(log(x))^2) + (4*x ^2*log(5)^2*log(x))/(3*log(log(x)))
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx=\frac {\mathrm {log}\left (5\right )^{2} \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} \mathrm {log}\left (x \right )^{2}+8 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} \mathrm {log}\left (x \right ) x +16 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) x^{2}+32 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{3}+16 x^{4}\right )}{6 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}} \] Input:
int(1/3*(((1+4*x)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log( x))^3+(8*x^2*log(5)^2*log(x)^2+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x)) ^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16*x^4*log (5)^2)/x/log(x)/log(log(x))^3,x)
Output:
(log(5)**2*(log(log(x))**2*log(x)**2 + 8*log(log(x))**2*log(x)*x + 16*log( log(x))**2*x**2 + 8*log(log(x))*log(x)*x**2 + 32*log(log(x))*x**3 + 16*x** 4))/(6*log(log(x))**2)