Integrand size = 199, antiderivative size = 28 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{x \left (x+4 x^2\right ) (x+\log (4-7 x-\log (x)))} \]
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{x^2 (1+4 x) (x+\log (4-7 x-\log (x)))} \]
Integrate[(-4 + 4*x + 60*x^2 - 448*x^3 + (-12*x - 64*x^2)*Log[x] + (32 + 1 36*x - 336*x^2 + (-8 - 48*x)*Log[x])*Log[4 - 7*x - Log[x]])/(-4*x^5 - 25*x ^6 - 8*x^7 + 112*x^8 + (x^5 + 8*x^6 + 16*x^7)*Log[x] + (-8*x^4 - 50*x^5 - 16*x^6 + 224*x^7 + (2*x^4 + 16*x^5 + 32*x^6)*Log[x])*Log[4 - 7*x - Log[x]] + (-4*x^3 - 25*x^4 - 8*x^5 + 112*x^6 + (x^3 + 8*x^4 + 16*x^5)*Log[x])*Log [4 - 7*x - 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.
| \(\displaystyle \int \frac {-448 x^3+60 x^2+\left (-64 x^2-12 x\right ) \log (x)+\left (-336 x^2+136 x+(-48 x-8) \log (x)+32\right ) \log (-7 x-\log (x)+4)+4 x-4}{112 x^8-8 x^7-25 x^6-4 x^5+\left (16 x^7+8 x^6+x^5\right ) \log (x)+\left (224 x^7-16 x^6-50 x^5-8 x^4+\left (32 x^6+16 x^5+2 x^4\right ) \log (x)\right ) \log (-7 x-\log (x)+4)+\left (112 x^6-8 x^5-25 x^4-4 x^3+\left (16 x^5+8 x^4+x^3\right ) \log (x)\right ) \log ^2(-7 x-\log (x)+4)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (112 x^3-15 x^2+\left (84 x^2-34 x-8\right ) \log (-7 x-\log (x)+4)-x+\log (x) (x (16 x+3)+2 (6 x+1) \log (-7 x-\log (x)+4))+1\right )}{x^3 (4 x+1)^2 (-7 x-\log (x)+4) (x+\log (-7 x-\log (x)+4))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {112 x^3-15 x^2-x-2 \left (-42 x^2+17 x+4\right ) \log (-7 x-\log (x)+4)+\log (x) (x (16 x+3)+2 (6 x+1) \log (-7 x-\log (x)+4))+1}{x^3 (4 x+1)^2 (-7 x-\log (x)+4) (x+\log (-7 x-\log (x)+4))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (\frac {-7 x^2-\log (x) x-3 x-1}{x^3 (4 x+1) (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}-\frac {2 (6 x+1)}{x^3 (4 x+1)^2 (x+\log (-7 x-\log (x)+4))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \left (-\int \frac {1}{x^3 (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx-2 \int \frac {1}{x^3 (x+\log (-7 x-\log (x)+4))}dx+\int \frac {1}{x^2 (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx-\int \frac {\log (x)}{x^2 (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+4 \int \frac {1}{x^2 (x+\log (-7 x-\log (x)+4))}dx+48 \int \frac {1}{(7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+5 \int \frac {1}{x (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+112 \int \frac {x}{(7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+44 \int \frac {1}{(4 x+1) (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+16 \int \frac {\log (x)}{(7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx+4 \int \frac {\log (x)}{x (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx-16 \int \frac {\log (x)}{(4 x+1) (7 x+\log (x)-4) (x+\log (-7 x-\log (x)+4))^2}dx-64 \int \frac {1}{(4 x+1)^2 (x+\log (-7 x-\log (x)+4))}dx+\frac {16}{x+\log (-7 x-\log (x)+4)}\right )\) |
Int[(-4 + 4*x + 60*x^2 - 448*x^3 + (-12*x - 64*x^2)*Log[x] + (32 + 136*x - 336*x^2 + (-8 - 48*x)*Log[x])*Log[4 - 7*x - Log[x]])/(-4*x^5 - 25*x^6 - 8 *x^7 + 112*x^8 + (x^5 + 8*x^6 + 16*x^7)*Log[x] + (-8*x^4 - 50*x^5 - 16*x^6 + 224*x^7 + (2*x^4 + 16*x^5 + 32*x^6)*Log[x])*Log[4 - 7*x - Log[x]] + (-4 *x^3 - 25*x^4 - 8*x^5 + 112*x^6 + (x^3 + 8*x^4 + 16*x^5)*Log[x])*Log[4 - 7 *x - Log[x]]^2),x]
3.8.96.3.1 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 = 5.82 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {4}{x^{2} \left (1+4 x \right ) \left (x +\ln \left (-\ln \left (x \right )-7 x +4\right )\right )}\) | \(27\) |
| risch | \(\frac {4}{x^{2} \left (1+4 x \right ) \left (x +\ln \left (-\ln \left (x \right )-7 x +4\right )\right )}\) | \(27\) |
| parallelrisch | \(\frac {4}{x^{2} \left (4 x \ln \left (-\ln \left (x \right )-7 x +4\right )+4 x^{2}+\ln \left (-\ln \left (x \right )-7 x +4\right )+x \right )}\) | \(38\) |
int((((-48*x-8)*ln(x)-336*x^2+136*x+32)*ln(-ln(x)-7*x+4)+(-64*x^2-12*x)*ln (x)-448*x^3+60*x^2+4*x-4)/(((16*x^5+8*x^4+x^3)*ln(x)+112*x^6-8*x^5-25*x^4- 4*x^3)*ln(-ln(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*ln(x)+224*x^7-16*x^6-50*x ^5-8*x^4)*ln(-ln(x)-7*x+4)+(16*x^7+8*x^6+x^5)*ln(x)+112*x^8-8*x^7-25*x^6-4 *x^5),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{4 \, x^{4} + x^{3} + {\left (4 \, x^{3} + x^{2}\right )} \log \left (-7 \, x - \log \left (x\right ) + 4\right )} \]
integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2 -12*x)*log(x)-448*x^3+60*x^2+4*x-4)/(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8* x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224*x ^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x ^8-8*x^7-25*x^6-4*x^5),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{4 x^{4} + x^{3} + \left (4 x^{3} + x^{2}\right ) \log {\left (- 7 x - \log {\left (x \right )} + 4 \right )}} \]
integrate((((-48*x-8)*ln(x)-336*x**2+136*x+32)*ln(-ln(x)-7*x+4)+(-64*x**2- 12*x)*ln(x)-448*x**3+60*x**2+4*x-4)/(((16*x**5+8*x**4+x**3)*ln(x)+112*x**6 -8*x**5-25*x**4-4*x**3)*ln(-ln(x)-7*x+4)**2+((32*x**6+16*x**5+2*x**4)*ln(x )+224*x**7-16*x**6-50*x**5-8*x**4)*ln(-ln(x)-7*x+4)+(16*x**7+8*x**6+x**5)* ln(x)+112*x**8-8*x**7-25*x**6-4*x**5),x)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{4 \, x^{4} + x^{3} + {\left (4 \, x^{3} + x^{2}\right )} \log \left (-7 \, x - \log \left (x\right ) + 4\right )} \]
integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2 -12*x)*log(x)-448*x^3+60*x^2+4*x-4)/(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8* x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224*x ^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x ^8-8*x^7-25*x^6-4*x^5),x, algorithm=\
Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{4 \, x^{4} + 4 \, x^{3} \log \left (-7 \, x - \log \left (x\right ) + 4\right ) + x^{3} + x^{2} \log \left (-7 \, x - \log \left (x\right ) + 4\right )} \]
integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2 -12*x)*log(x)-448*x^3+60*x^2+4*x-4)/(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8* x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224*x ^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x ^8-8*x^7-25*x^6-4*x^5),x, algorithm=\
Time = 9.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-4+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx=\frac {4}{x^2\,\left (4\,x+1\right )\,\left (x+\ln \left (4-\ln \left (x\right )-7\,x\right )\right )} \]
int(-(4*x + log(4 - log(x) - 7*x)*(136*x - log(x)*(48*x + 8) - 336*x^2 + 3 2) - log(x)*(12*x + 64*x^2) + 60*x^2 - 448*x^3 - 4)/(log(4 - log(x) - 7*x) *(8*x^4 - log(x)*(2*x^4 + 16*x^5 + 32*x^6) + 50*x^5 + 16*x^6 - 224*x^7) + log(4 - log(x) - 7*x)^2*(4*x^3 + 25*x^4 + 8*x^5 - 112*x^6 - log(x)*(x^3 + 8*x^4 + 16*x^5)) + 4*x^5 + 25*x^6 + 8*x^7 - 112*x^8 - log(x)*(x^5 + 8*x^6 + 16*x^7)),x)