Integrand size = 261, antiderivative size = 28 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=\frac {5}{2 \left (1+\frac {x}{2}\right ) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))} \]
Time = 0.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=\frac {5}{(2+x) \left (x+\log ^2(x)\right ) (-4+\log (\log (x)))} \]
Integrate[(-10*x - 5*x^2 + (40*x + 40*x^2)*Log[x] + (70 + 35*x)*Log[x]^2 + 20*x*Log[x]^3 + ((-10*x - 10*x^2)*Log[x] + (-20 - 10*x)*Log[x]^2 - 5*x*Lo g[x]^3)*Log[Log[x]])/((64*x^3 + 64*x^4 + 16*x^5)*Log[x] + (128*x^2 + 128*x ^3 + 32*x^4)*Log[x]^3 + (64*x + 64*x^2 + 16*x^3)*Log[x]^5 + ((-32*x^3 - 32 *x^4 - 8*x^5)*Log[x] + (-64*x^2 - 64*x^3 - 16*x^4)*Log[x]^3 + (-32*x - 32* x^2 - 8*x^3)*Log[x]^5)*Log[Log[x]] + ((4*x^3 + 4*x^4 + x^5)*Log[x] + (8*x^ 2 + 8*x^3 + 2*x^4)*Log[x]^3 + (4*x + 4*x^2 + x^3)*Log[x]^5)*Log[Log[x]]^2) ,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.
| \(\displaystyle \int \frac {-5 x^2+\left (\left (-10 x^2-10 x\right ) \log (x)-5 x \log ^3(x)+(-10 x-20) \log ^2(x)\right ) \log (\log (x))+\left (40 x^2+40 x\right ) \log (x)-10 x+20 x \log ^3(x)+(35 x+70) \log ^2(x)}{\left (16 x^3+64 x^2+64 x\right ) \log ^5(x)+\left (16 x^5+64 x^4+64 x^3\right ) \log (x)+\left (32 x^4+128 x^3+128 x^2\right ) \log ^3(x)+\left (\left (-8 x^3-32 x^2-32 x\right ) \log ^5(x)+\left (-8 x^5-32 x^4-32 x^3\right ) \log (x)+\left (-16 x^4-64 x^3-64 x^2\right ) \log ^3(x)\right ) \log (\log (x))+\left (\left (x^3+4 x^2+4 x\right ) \log ^5(x)+\left (x^5+4 x^4+4 x^3\right ) \log (x)+\left (2 x^4+8 x^3+8 x^2\right ) \log ^3(x)\right ) \log ^2(\log (x))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (-x (x+2)-x (\log (\log (x))-4) \log ^3(x)-(x+2) (2 \log (\log (x))-7) \log ^2(x)-2 x (x+1) (\log (\log (x))-4) \log (x)\right )}{x (x+2)^2 \log (x) \left (x+\log ^2(x)\right )^2 (4-\log (\log (x)))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {-x (4-\log (\log (x))) \log ^3(x)-(x+2) (7-2 \log (\log (x))) \log ^2(x)-2 x (x+1) (4-\log (\log (x))) \log (x)+x (x+2)}{x (x+2)^2 \log (x) \left (\log ^2(x)+x\right )^2 (4-\log (\log (x)))^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {-x (4-\log (\log (x))) \log ^3(x)-(x+2) (7-2 \log (\log (x))) \log ^2(x)-2 x (x+1) (4-\log (\log (x))) \log (x)+x (x+2)}{x (x+2)^2 \log (x) \left (\log ^2(x)+x\right )^2 (4-\log (\log (x)))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {2 x^2+\log ^2(x) x+2 \log (x) x+2 x+4 \log (x)}{x (x+2)^2 \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}+\frac {1}{x (x+2) \log (x) \left (\log ^2(x)+x\right ) (\log (\log (x))-4)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \left (\frac {1}{2} \int \frac {1}{x \log (x) \left (\log ^2(x)+x\right ) (\log (\log (x))-4)^2}dx-\frac {1}{2} \int \frac {1}{(x+2) \log (x) \left (\log ^2(x)+x\right ) (\log (\log (x))-4)^2}dx-2 \int \frac {1}{(x+2)^2 \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}dx+2 \int \frac {1}{(x+2) \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}dx+\int \frac {\log (x)}{x \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}dx-\int \frac {\log (x)}{(x+2) \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}dx+\int \frac {\log ^2(x)}{(x+2)^2 \left (\log ^2(x)+x\right )^2 (\log (\log (x))-4)}dx\right )\) |
Int[(-10*x - 5*x^2 + (40*x + 40*x^2)*Log[x] + (70 + 35*x)*Log[x]^2 + 20*x* Log[x]^3 + ((-10*x - 10*x^2)*Log[x] + (-20 - 10*x)*Log[x]^2 - 5*x*Log[x]^3 )*Log[Log[x]])/((64*x^3 + 64*x^4 + 16*x^5)*Log[x] + (128*x^2 + 128*x^3 + 3 2*x^4)*Log[x]^3 + (64*x + 64*x^2 + 16*x^3)*Log[x]^5 + ((-32*x^3 - 32*x^4 - 8*x^5)*Log[x] + (-64*x^2 - 64*x^3 - 16*x^4)*Log[x]^3 + (-32*x - 32*x^2 - 8*x^3)*Log[x]^5)*Log[Log[x]] + ((4*x^3 + 4*x^4 + x^5)*Log[x] + (8*x^2 + 8* x^3 + 2*x^4)*Log[x]^3 + (4*x + 4*x^2 + x^3)*Log[x]^5)*Log[Log[x]]^2),x]
3.10.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 13.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(\frac {5}{\left (x \ln \left (x \right )^{2}+2 \ln \left (x \right )^{2}+x^{2}+2 x \right ) \left (\ln \left (\ln \left (x \right )\right )-4\right )}\) | \(31\) |
| risch | \(\frac {5}{\left (x \ln \left (x \right )^{2}+2 \ln \left (x \right )^{2}+x^{2}+2 x \right ) \left (\ln \left (\ln \left (x \right )\right )-4\right )}\) | \(31\) |
| parallelrisch | \(\frac {5}{\left (x \ln \left (x \right )^{2}+2 \ln \left (x \right )^{2}+x^{2}+2 x \right ) \left (\ln \left (\ln \left (x \right )\right )-4\right )}\) | \(31\) |
int(((-5*x*ln(x)^3+(-10*x-20)*ln(x)^2+(-10*x^2-10*x)*ln(x))*ln(ln(x))+20*x *ln(x)^3+(35*x+70)*ln(x)^2+(40*x^2+40*x)*ln(x)-5*x^2-10*x)/(((x^3+4*x^2+4* x)*ln(x)^5+(2*x^4+8*x^3+8*x^2)*ln(x)^3+(x^5+4*x^4+4*x^3)*ln(x))*ln(ln(x))^ 2+((-8*x^3-32*x^2-32*x)*ln(x)^5+(-16*x^4-64*x^3-64*x^2)*ln(x)^3+(-8*x^5-32 *x^4-32*x^3)*ln(x))*ln(ln(x))+(16*x^3+64*x^2+64*x)*ln(x)^5+(32*x^4+128*x^3 +128*x^2)*ln(x)^3+(16*x^5+64*x^4+64*x^3)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=-\frac {5}{4 \, {\left (x + 2\right )} \log \left (x\right )^{2} + 4 \, x^{2} - {\left ({\left (x + 2\right )} \log \left (x\right )^{2} + x^{2} + 2 \, x\right )} \log \left (\log \left (x\right )\right ) + 8 \, x} \]
integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(l og(x))+20*x*log(x)^3+(35*x+70)*log(x)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/( ((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)*l og(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2 )*log(x)^3+(-8*x^5-32*x^4-32*x^3)*log(x))*log(log(x))+(16*x^3+64*x^2+64*x) *log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log(x)) ,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=\frac {5}{- 4 x^{2} - 4 x \log {\left (x \right )}^{2} - 8 x + \left (x^{2} + x \log {\left (x \right )}^{2} + 2 x + 2 \log {\left (x \right )}^{2}\right ) \log {\left (\log {\left (x \right )} \right )} - 8 \log {\left (x \right )}^{2}} \]
integrate(((-5*x*ln(x)**3+(-10*x-20)*ln(x)**2+(-10*x**2-10*x)*ln(x))*ln(ln (x))+20*x*ln(x)**3+(35*x+70)*ln(x)**2+(40*x**2+40*x)*ln(x)-5*x**2-10*x)/(( (x**3+4*x**2+4*x)*ln(x)**5+(2*x**4+8*x**3+8*x**2)*ln(x)**3+(x**5+4*x**4+4* x**3)*ln(x))*ln(ln(x))**2+((-8*x**3-32*x**2-32*x)*ln(x)**5+(-16*x**4-64*x* *3-64*x**2)*ln(x)**3+(-8*x**5-32*x**4-32*x**3)*ln(x))*ln(ln(x))+(16*x**3+6 4*x**2+64*x)*ln(x)**5+(32*x**4+128*x**3+128*x**2)*ln(x)**3+(16*x**5+64*x** 4+64*x**3)*ln(x)),x)
5/(-4*x**2 - 4*x*log(x)**2 - 8*x + (x**2 + x*log(x)**2 + 2*x + 2*log(x)**2 )*log(log(x)) - 8*log(x)**2)
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=-\frac {5}{4 \, {\left (x + 2\right )} \log \left (x\right )^{2} + 4 \, x^{2} - {\left ({\left (x + 2\right )} \log \left (x\right )^{2} + x^{2} + 2 \, x\right )} \log \left (\log \left (x\right )\right ) + 8 \, x} \]
integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(l og(x))+20*x*log(x)^3+(35*x+70)*log(x)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/( ((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)*l og(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2 )*log(x)^3+(-8*x^5-32*x^4-32*x^3)*log(x))*log(log(x))+(16*x^3+64*x^2+64*x) *log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log(x)) ,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).
Time = 0.49 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=\frac {5}{x \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) - 4 \, x \log \left (x\right )^{2} + x^{2} \log \left (\log \left (x\right )\right ) + 2 \, \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) - 4 \, x^{2} - 8 \, \log \left (x\right )^{2} + 2 \, x \log \left (\log \left (x\right )\right ) - 8 \, x} \]
integrate(((-5*x*log(x)^3+(-10*x-20)*log(x)^2+(-10*x^2-10*x)*log(x))*log(l og(x))+20*x*log(x)^3+(35*x+70)*log(x)^2+(40*x^2+40*x)*log(x)-5*x^2-10*x)/( ((x^3+4*x^2+4*x)*log(x)^5+(2*x^4+8*x^3+8*x^2)*log(x)^3+(x^5+4*x^4+4*x^3)*l og(x))*log(log(x))^2+((-8*x^3-32*x^2-32*x)*log(x)^5+(-16*x^4-64*x^3-64*x^2 )*log(x)^3+(-8*x^5-32*x^4-32*x^3)*log(x))*log(log(x))+(16*x^3+64*x^2+64*x) *log(x)^5+(32*x^4+128*x^3+128*x^2)*log(x)^3+(16*x^5+64*x^4+64*x^3)*log(x)) ,x, algorithm=\
5/(x*log(x)^2*log(log(x)) - 4*x*log(x)^2 + x^2*log(log(x)) + 2*log(x)^2*lo g(log(x)) - 4*x^2 - 8*log(x)^2 + 2*x*log(log(x)) - 8*x)
Time = 9.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-10 x-5 x^2+\left (40 x+40 x^2\right ) \log (x)+(70+35 x) \log ^2(x)+20 x \log ^3(x)+\left (\left (-10 x-10 x^2\right ) \log (x)+(-20-10 x) \log ^2(x)-5 x \log ^3(x)\right ) \log (\log (x))}{\left (64 x^3+64 x^4+16 x^5\right ) \log (x)+\left (128 x^2+128 x^3+32 x^4\right ) \log ^3(x)+\left (64 x+64 x^2+16 x^3\right ) \log ^5(x)+\left (\left (-32 x^3-32 x^4-8 x^5\right ) \log (x)+\left (-64 x^2-64 x^3-16 x^4\right ) \log ^3(x)+\left (-32 x-32 x^2-8 x^3\right ) \log ^5(x)\right ) \log (\log (x))+\left (\left (4 x^3+4 x^4+x^5\right ) \log (x)+\left (8 x^2+8 x^3+2 x^4\right ) \log ^3(x)+\left (4 x+4 x^2+x^3\right ) \log ^5(x)\right ) \log ^2(\log (x))} \, dx=\frac {5}{\left ({\ln \left (x\right )}^2+x\right )\,\left (\ln \left (\ln \left (x\right )\right )-4\right )\,\left (x+2\right )} \]
int(-(10*x - 20*x*log(x)^3 - log(x)*(40*x + 40*x^2) + log(log(x))*(5*x*log (x)^3 + log(x)*(10*x + 10*x^2) + log(x)^2*(10*x + 20)) + 5*x^2 - log(x)^2* (35*x + 70))/(log(x)^5*(64*x + 64*x^2 + 16*x^3) - log(log(x))*(log(x)^5*(3 2*x + 32*x^2 + 8*x^3) + log(x)*(32*x^3 + 32*x^4 + 8*x^5) + log(x)^3*(64*x^ 2 + 64*x^3 + 16*x^4)) + log(x)*(64*x^3 + 64*x^4 + 16*x^5) + log(x)^3*(128* x^2 + 128*x^3 + 32*x^4) + log(log(x))^2*(log(x)^3*(8*x^2 + 8*x^3 + 2*x^4) + log(x)^5*(4*x + 4*x^2 + x^3) + log(x)*(4*x^3 + 4*x^4 + x^5))),x)