Integrand size = 166, antiderivative size = 31 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=\frac {x}{2}-x^2 \left (x+\log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )\right )^2 \] Output:
1/2*x-(ln(x^2/(ln(5)+x)^2/ln(ln(x)))+x)^2*x^2
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=\frac {x}{2}-x^4-2 x^3 \log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )-x^2 \log ^2\left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right ) \] Input:
Integrate[(4*x^3 + 4*x^2*Log[5] + (x - 8*x^4 + (1 - 8*x^2 - 8*x^3)*Log[5]) *Log[x]*Log[Log[x]] + (4*x^2 + 4*x*Log[5] + (-12*x^3 + (-8*x - 12*x^2)*Log [5])*Log[x]*Log[Log[x]])*Log[x^2/((x^2 + 2*x*Log[5] + Log[5]^2)*Log[Log[x] ])] + (-4*x^2 - 4*x*Log[5])*Log[x]*Log[Log[x]]*Log[x^2/((x^2 + 2*x*Log[5] + Log[5]^2)*Log[Log[x]])]^2)/((2*x + 2*Log[5])*Log[x]*Log[Log[x]]),x]
Output:
x/2 - x^4 - 2*x^3*Log[x^2/((x + Log[5])^2*Log[Log[x]])] - x^2*Log[x^2/((x + Log[5])^2*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.
| \(\displaystyle \int \frac {4 x^3+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+4 x^2 \log (5)+\left (4 x^2+\left (\left (-12 x^2-8 x\right ) \log (5)-12 x^3\right ) \log (x) \log (\log (x))+4 x \log (5)\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-8 x^4+\left (-8 x^3-8 x^2+1\right ) \log (5)+x\right ) \log (x) \log (\log (x))}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x \log ^2\left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )-\frac {2 x \left (3 x^2 \log (x) \log (\log (x))-x+3 x \log (5) \log (x) \log (\log (x))+\log (25) \log (x) \log (\log (x))-\log (5)\right ) \log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )}{(x+\log (5)) \log (x) \log (\log (x))}+\frac {-8 x^4 \log (x) \log (\log (x))+4 x^3-8 x^3 \log (5) \log (x) \log (\log (x))-8 x^2 \log (5) \log (x) \log (\log (x))+4 x^2 \log (5)+x \log (x) \log (\log (x))+\log (5) \log (x) \log (\log (x))}{2 (x+\log (5)) \log (x) \log (\log (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int x \log ^2\left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )dx+2 \log (5) \log (25) \int \frac {\log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )}{x+\log (5)}dx+2 \int \frac {x \log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )}{\log (x) \log (\log (x))}dx-2 \log (25) \int \frac {1}{\log (x) \log (\log (x))}dx-x^4+x^2 \log (25)-2 x^2 \log (5)-2 x \log (25) \log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )-2 x^3 \log \left (\frac {x^2}{(x+\log (5))^2 \log (\log (x))}\right )-4 \log ^3(5) \log (x+\log (5))+\frac {1}{2} x \left (1+8 \log ^2(5)\right )+2 \log ^2(25) \log (x+\log (5))+2 \log ^2(5) \log (25) \log (x+\log (5))-2 x \log (5) \log (25)\) |
Input:
Int[(4*x^3 + 4*x^2*Log[5] + (x - 8*x^4 + (1 - 8*x^2 - 8*x^3)*Log[5])*Log[x ]*Log[Log[x]] + (4*x^2 + 4*x*Log[5] + (-12*x^3 + (-8*x - 12*x^2)*Log[5])*L og[x]*Log[Log[x]])*Log[x^2/((x^2 + 2*x*Log[5] + Log[5]^2)*Log[Log[x]])] + (-4*x^2 - 4*x*Log[5])*Log[x]*Log[Log[x]]*Log[x^2/((x^2 + 2*x*Log[5] + Log[ 5]^2)*Log[Log[x]])]^2)/((2*x + 2*Log[5])*Log[x]*Log[Log[x]]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(29)=58\).
Time = 34.66 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45
| method | result | size |
| parallelrisch | \(-x^{4}-2 \ln \left (\frac {x^{2}}{\left (\ln \left (5\right )^{2}+2 x \ln \left (5\right )+x^{2}\right ) \ln \left (\ln \left (x \right )\right )}\right ) x^{3}-\ln \left (\frac {x^{2}}{\left (\ln \left (5\right )^{2}+2 x \ln \left (5\right )+x^{2}\right ) \ln \left (\ln \left (x \right )\right )}\right )^{2} x^{2}-\ln \left (5\right )+\frac {x}{2}\) | \(76\) |
| risch | \(\text {Expression too large to display}\) | \(6484\) |
Input:
int(((-4*x*ln(5)-4*x^2)*ln(x)*ln(ln(x))*ln(x^2/(ln(5)^2+2*x*ln(5)+x^2)/ln( ln(x)))^2+(((-12*x^2-8*x)*ln(5)-12*x^3)*ln(x)*ln(ln(x))+4*x*ln(5)+4*x^2)*l n(x^2/(ln(5)^2+2*x*ln(5)+x^2)/ln(ln(x)))+((-8*x^3-8*x^2+1)*ln(5)-8*x^4+x)* ln(x)*ln(ln(x))+4*x^2*ln(5)+4*x^3)/(2*ln(5)+2*x)/ln(x)/ln(ln(x)),x,method= _RETURNVERBOSE)
Output:
-x^4-2*ln(x^2/(ln(5)^2+2*x*ln(5)+x^2)/ln(ln(x)))*x^3-ln(x^2/(ln(5)^2+2*x*l n(5)+x^2)/ln(ln(x)))^2*x^2-ln(5)+1/2*x
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=-x^{4} - 2 \, x^{3} \log \left (\frac {x^{2}}{{\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} \log \left (\log \left (x\right )\right )}\right ) - x^{2} \log \left (\frac {x^{2}}{{\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} \log \left (\log \left (x\right )\right )}\right )^{2} + \frac {1}{2} \, x \] Input:
integrate(((-4*x*log(5)-4*x^2)*log(x)*log(log(x))*log(x^2/(log(5)^2+2*x*lo g(5)+x^2)/log(log(x)))^2+(((-12*x^2-8*x)*log(5)-12*x^3)*log(x)*log(log(x)) +4*x*log(5)+4*x^2)*log(x^2/(log(5)^2+2*x*log(5)+x^2)/log(log(x)))+((-8*x^3 -8*x^2+1)*log(5)-8*x^4+x)*log(x)*log(log(x))+4*x^2*log(5)+4*x^3)/(2*log(5) +2*x)/log(x)/log(log(x)),x, algorithm="fricas")
Output:
-x^4 - 2*x^3*log(x^2/((x^2 + 2*x*log(5) + log(5)^2)*log(log(x)))) - x^2*lo g(x^2/((x^2 + 2*x*log(5) + log(5)^2)*log(log(x))))^2 + 1/2*x
Exception generated. \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-4*x*ln(5)-4*x**2)*ln(x)*ln(ln(x))*ln(x**2/(ln(5)**2+2*x*ln(5) +x**2)/ln(ln(x)))**2+(((-12*x**2-8*x)*ln(5)-12*x**3)*ln(x)*ln(ln(x))+4*x*l n(5)+4*x**2)*ln(x**2/(ln(5)**2+2*x*ln(5)+x**2)/ln(ln(x)))+((-8*x**3-8*x**2 +1)*ln(5)-8*x**4+x)*ln(x)*ln(ln(x))+4*x**2*ln(5)+4*x**3)/(2*ln(5)+2*x)/ln( x)/ln(ln(x)),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (29) = 58\).
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.97 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=-x^{4} - 4 \, x^{2} \log \left (x + \log \left (5\right )\right )^{2} - 4 \, x^{3} \log \left (x\right ) - 4 \, x^{2} \log \left (x\right )^{2} - x^{2} \log \left (\log \left (\log \left (x\right )\right )\right )^{2} + 4 \, {\left (x^{3} + 2 \, x^{2} \log \left (x\right ) - x^{2} \log \left (\log \left (\log \left (x\right )\right )\right )\right )} \log \left (x + \log \left (5\right )\right ) + 2 \, {\left (x^{3} + 2 \, x^{2} \log \left (x\right )\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) + \frac {1}{2} \, x \] Input:
integrate(((-4*x*log(5)-4*x^2)*log(x)*log(log(x))*log(x^2/(log(5)^2+2*x*lo g(5)+x^2)/log(log(x)))^2+(((-12*x^2-8*x)*log(5)-12*x^3)*log(x)*log(log(x)) +4*x*log(5)+4*x^2)*log(x^2/(log(5)^2+2*x*log(5)+x^2)/log(log(x)))+((-8*x^3 -8*x^2+1)*log(5)-8*x^4+x)*log(x)*log(log(x))+4*x^2*log(5)+4*x^3)/(2*log(5) +2*x)/log(x)/log(log(x)),x, algorithm="maxima")
Output:
-x^4 - 4*x^2*log(x + log(5))^2 - 4*x^3*log(x) - 4*x^2*log(x)^2 - x^2*log(l og(log(x)))^2 + 4*(x^3 + 2*x^2*log(x) - x^2*log(log(log(x))))*log(x + log( 5)) + 2*(x^3 + 2*x^2*log(x))*log(log(log(x))) + 1/2*x
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (29) = 58\).
Time = 2.88 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=-x^{4} - x^{2} \log \left (x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x \log \left (5\right ) \log \left (\log \left (x\right )\right ) + \log \left (5\right )^{2} \log \left (\log \left (x\right )\right )\right )^{2} - 4 \, x^{3} \log \left (x\right ) - 4 \, x^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{3} + 2 \, x^{2} \log \left (x\right )\right )} \log \left (x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x \log \left (5\right ) \log \left (\log \left (x\right )\right ) + \log \left (5\right )^{2} \log \left (\log \left (x\right )\right )\right ) + \frac {1}{2} \, x \] Input:
integrate(((-4*x*log(5)-4*x^2)*log(x)*log(log(x))*log(x^2/(log(5)^2+2*x*lo g(5)+x^2)/log(log(x)))^2+(((-12*x^2-8*x)*log(5)-12*x^3)*log(x)*log(log(x)) +4*x*log(5)+4*x^2)*log(x^2/(log(5)^2+2*x*log(5)+x^2)/log(log(x)))+((-8*x^3 -8*x^2+1)*log(5)-8*x^4+x)*log(x)*log(log(x))+4*x^2*log(5)+4*x^3)/(2*log(5) +2*x)/log(x)/log(log(x)),x, algorithm="giac")
Output:
-x^4 - x^2*log(x^2*log(log(x)) + 2*x*log(5)*log(log(x)) + log(5)^2*log(log (x)))^2 - 4*x^3*log(x) - 4*x^2*log(x)^2 + 2*(x^3 + 2*x^2*log(x))*log(x^2*l og(log(x)) + 2*x*log(5)*log(log(x)) + log(5)^2*log(log(x))) + 1/2*x
Timed out. \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=\int \frac {\ln \left (\frac {x^2}{\ln \left (\ln \left (x\right )\right )\,\left (x^2+2\,\ln \left (5\right )\,x+{\ln \left (5\right )}^2\right )}\right )\,\left (4\,x\,\ln \left (5\right )+4\,x^2-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (\ln \left (5\right )\,\left (12\,x^2+8\,x\right )+12\,x^3\right )\right )+4\,x^2\,\ln \left (5\right )+4\,x^3-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (\ln \left (5\right )\,\left (8\,x^3+8\,x^2-1\right )-x+8\,x^4\right )-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,{\ln \left (\frac {x^2}{\ln \left (\ln \left (x\right )\right )\,\left (x^2+2\,\ln \left (5\right )\,x+{\ln \left (5\right )}^2\right )}\right )}^2\,\left (4\,x^2+4\,\ln \left (5\right )\,x\right )}{\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x+2\,\ln \left (5\right )\right )} \,d x \] Input:
int((log(x^2/(log(log(x))*(2*x*log(5) + log(5)^2 + x^2)))*(4*x*log(5) + 4* x^2 - log(log(x))*log(x)*(log(5)*(8*x + 12*x^2) + 12*x^3)) + 4*x^2*log(5) + 4*x^3 - log(log(x))*log(x)*(log(5)*(8*x^2 + 8*x^3 - 1) - x + 8*x^4) - lo g(log(x))*log(x)*log(x^2/(log(log(x))*(2*x*log(5) + log(5)^2 + x^2)))^2*(4 *x*log(5) + 4*x^2))/(log(log(x))*log(x)*(2*x + 2*log(5))),x)
Output:
int((log(x^2/(log(log(x))*(2*x*log(5) + log(5)^2 + x^2)))*(4*x*log(5) + 4* x^2 - log(log(x))*log(x)*(log(5)*(8*x + 12*x^2) + 12*x^3)) + 4*x^2*log(5) + 4*x^3 - log(log(x))*log(x)*(log(5)*(8*x^2 + 8*x^3 - 1) - x + 8*x^4) - lo g(log(x))*log(x)*log(x^2/(log(log(x))*(2*x*log(5) + log(5)^2 + x^2)))^2*(4 *x*log(5) + 4*x^2))/(log(log(x))*log(x)*(2*x + 2*log(5))), x)
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65 \[ \int \frac {4 x^3+4 x^2 \log (5)+\left (x-8 x^4+\left (1-8 x^2-8 x^3\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (4 x^2+4 x \log (5)+\left (-12 x^3+\left (-8 x-12 x^2\right ) \log (5)\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )+\left (-4 x^2-4 x \log (5)\right ) \log (x) \log (\log (x)) \log ^2\left (\frac {x^2}{\left (x^2+2 x \log (5)+\log ^2(5)\right ) \log (\log (x))}\right )}{(2 x+2 \log (5)) \log (x) \log (\log (x))} \, dx=\frac {x \left (-2 \mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right ) x +\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}}\right )^{2} x -4 \,\mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right )^{2}+2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right ) x +\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}}\right ) x^{2}-2 x^{3}+1\right )}{2} \] Input:
int(((-4*x*log(5)-4*x^2)*log(x)*log(log(x))*log(x^2/(log(5)^2+2*x*log(5)+x ^2)/log(log(x)))^2+(((-12*x^2-8*x)*log(5)-12*x^3)*log(x)*log(log(x))+4*x*l og(5)+4*x^2)*log(x^2/(log(5)^2+2*x*log(5)+x^2)/log(log(x)))+((-8*x^3-8*x^2 +1)*log(5)-8*x^4+x)*log(x)*log(log(x))+4*x^2*log(5)+4*x^3)/(2*log(5)+2*x)/ log(x)/log(log(x)),x)
Output:
(x*( - 2*log(x**2/(log(log(x))*log(5)**2 + 2*log(log(x))*log(5)*x + log(lo g(x))*x**2))**2*x - 4*log(x**2/(log(log(x))*log(5)**2 + 2*log(log(x))*log( 5)*x + log(log(x))*x**2))*x**2 - 2*x**3 + 1))/2