Integrand size = 120, antiderivative size = 33 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=x-\frac {1}{9} \left (\frac {2}{x^2}-\log \left (e^{e^8+e^x} (x-\log (x))\right )\right )^2 \]
x-1/3*(2/x^2-ln((x-ln(x))*exp(exp(x)+exp(4)^2)))*(2/3/x^2-1/3*ln((x-ln(x)) *exp(exp(x)+exp(4)^2)))
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(33)=66\).
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=\frac {1}{9} \left (e^{2 x}-\frac {4}{x^4}+9 x+\log ^2(x-\log (x))+2 \log (x-\log (x)) \left (e^x-\log \left (e^{e^8+e^x} (x-\log (x))\right )\right )+\left (-2 e^x+\frac {4}{x^2}\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )\right ) \]
Integrate[(-16*x + 4*x^2 - 4*x^3 - 4*E^x*x^4 - 9*x^6 + (16 + 4*E^x*x^3 + 9 *x^5)*Log[x] + (8*x^3 - 2*x^4 + 2*x^5 + 2*E^x*x^6 + (-8*x^2 - 2*E^x*x^5)*L og[x])*Log[E^(E^8 + E^x)*(x - Log[x])])/(-9*x^6 + 9*x^5*Log[x]),x]
(E^(2*x) - 4/x^4 + 9*x + Log[x - Log[x]]^2 + 2*Log[x - Log[x]]*(E^x - Log[ E^(E^8 + E^x)*(x - Log[x])]) + (-2*E^x + 4/x^2)*Log[E^(E^8 + E^x)*(x - Log [x])])/9
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.
| \(\displaystyle \int \frac {-9 x^6-4 e^x x^4-4 x^3+4 x^2+\left (9 x^5+4 e^x x^3+16\right ) \log (x)+\left (2 e^x x^6+2 x^5-2 x^4+8 x^3+\left (-2 e^x x^5-8 x^2\right ) \log (x)\right ) \log \left (e^{e^x+e^8} (x-\log (x))\right )-16 x}{9 x^5 \log (x)-9 x^6} \, dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {-9 x^6-4 e^x x^4-4 x^3+4 x^2+\left (9 x^5+4 e^x x^3+16\right ) \log (x)+\left (2 e^x x^6+2 x^5-2 x^4+8 x^3+\left (-2 e^x x^5-8 x^2\right ) \log (x)\right ) \log \left (e^{e^x+e^8} (x-\log (x))\right )-16 x}{x^5 (9 \log (x)-9 x)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {9 x^6+4 e^x x^4+4 x^3-4 x^2-\left (9 x^5+4 e^x x^3+16\right ) \log (x)-\left (2 e^x x^6+2 x^5-2 x^4+8 x^3+\left (-2 e^x x^5-8 x^2\right ) \log (x)\right ) \log \left (e^{e^x+e^8} (x-\log (x))\right )+16 x}{9 x^5 (x-\log (x))}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {9 x^6+4 e^x x^4+4 x^3-4 x^2+16 x-\left (9 x^5+4 e^x x^3+16\right ) \log (x)-2 \left (e^x x^6+x^5-x^4+4 x^3-\left (e^x x^5+4 x^2\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{x^5 (x-\log (x))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{9} \int \left (\frac {9 x}{x-\log (x)}-\frac {9 \log (x)}{x-\log (x)}-\frac {2 \log \left (e^{e^8+e^x} (x-\log (x))\right )}{x-\log (x)}+\frac {2 \log \left (e^{e^8+e^x} (x-\log (x))\right )}{(x-\log (x)) x}-\frac {8 \log \left (e^{e^8+e^x} (x-\log (x))\right )}{(x-\log (x)) x^2}-\frac {2 e^x \left (x^2 \log \left (e^{e^8+e^x} (x-\log (x))\right )-2\right )}{x^2}+\frac {4}{(x-\log (x)) x^2}+\frac {8 \log (x) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{(x-\log (x)) x^3}-\frac {4}{(x-\log (x)) x^3}+\frac {16}{(x-\log (x)) x^4}-\frac {16 \log (x)}{(x-\log (x)) x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} \left (-4 \int \frac {1}{x^3 (x-\log (x))}dx+8 \int \frac {\log (x) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{x^3 (x-\log (x))}dx+4 \int \frac {1}{x^2 (x-\log (x))}dx-8 \int \frac {\log \left (e^{e^8+e^x} (x-\log (x))\right )}{x^2 (x-\log (x))}dx+2 \int \frac {e^x}{x-\log (x)}dx-2 \int \frac {e^x}{x (x-\log (x))}dx-2 \int \frac {\log \left (e^{e^8+e^x} (x-\log (x))\right )}{x-\log (x)}dx+2 \int \frac {\log \left (e^{e^8+e^x} (x-\log (x))\right )}{x (x-\log (x))}dx+4 \operatorname {ExpIntegralEi}(x)-\frac {4}{x^4}+9 x+e^{2 x}-\frac {4 e^x}{x}-2 e^x \log \left (e^{e^x+e^8} (x-\log (x))\right )\right )\) |
Int[(-16*x + 4*x^2 - 4*x^3 - 4*E^x*x^4 - 9*x^6 + (16 + 4*E^x*x^3 + 9*x^5)* Log[x] + (8*x^3 - 2*x^4 + 2*x^5 + 2*E^x*x^6 + (-8*x^2 - 2*E^x*x^5)*Log[x]) *Log[E^(E^8 + E^x)*(x - Log[x])])/(-9*x^6 + 9*x^5*Log[x]),x]
3.19.10.3.1 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_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.)) ^(p_.), x_Symbol] :> Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
Time = 28.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73
| method | result | size |
| parallelrisch | \(\frac {-8-2 \ln \left (\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2} x^{4}+18 x^{5}+8 x^{2} \ln \left (\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )}{18 x^{4}}\) | \(57\) |
| risch | \(-\frac {2 \left (\ln \left (x -\ln \left (x \right )\right ) x^{2}+{\mathrm e}^{x} x^{2}-2\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )}{9 x^{2}}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2} {\mathrm e}^{x} x^{4}-i \ln \left (\ln \left (x \right )-x \right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2} x^{4}-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2} {\mathrm e}^{x} x^{4}+2 i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{3}-2 i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) {\mathrm e}^{x} x^{4}+i \ln \left (\ln \left (x \right )-x \right ) \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) x^{4}-i \ln \left (\ln \left (x \right )-x \right ) \pi \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{3} x^{4}-i \pi \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{3} {\mathrm e}^{x} x^{4}-2 i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )+2 i \pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2}+i \ln \left (\ln \left (x \right )-x \right ) \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right )\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )-x \right ) {\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{8}}\right )^{2} x^{4}+{\mathrm e}^{2 x} x^{4}-\ln \left (x -\ln \left (x \right )\right )^{2} x^{4}+9 x^{5}+4 \ln \left (x -\ln \left (x \right )\right ) x^{2}-4}{9 x^{4}}\) | \(529\) |
int((((-2*x^5*exp(x)-8*x^2)*ln(x)+2*x^6*exp(x)+2*x^5-2*x^4+8*x^3)*ln((x-ln (x))*exp(exp(x)+exp(4)^2))+(4*exp(x)*x^3+9*x^5+16)*ln(x)-4*exp(x)*x^4-9*x^ 6-4*x^3+4*x^2-16*x)/(9*x^5*ln(x)-9*x^6),x,method=_RETURNVERBOSE)
1/18*(-8-2*ln((x-ln(x))*exp(exp(x)+exp(4)^2))^2*x^4+18*x^5+8*x^2*ln((x-ln( x))*exp(exp(x)+exp(4)^2)))/x^4
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=-\frac {x^{4} \log \left ({\left (x - \log \left (x\right )\right )} e^{\left (e^{8} + e^{x}\right )}\right )^{2} - 9 \, x^{5} - 4 \, x^{2} \log \left ({\left (x - \log \left (x\right )\right )} e^{\left (e^{8} + e^{x}\right )}\right ) + 4}{9 \, x^{4}} \]
integrate((((-2*x^5*exp(x)-8*x^2)*log(x)+2*x^6*exp(x)+2*x^5-2*x^4+8*x^3)*l og((x-log(x))*exp(exp(x)+exp(4)^2))+(4*exp(x)*x^3+9*x^5+16)*log(x)-4*exp(x )*x^4-9*x^6-4*x^3+4*x^2-16*x)/(9*x^5*log(x)-9*x^6),x, algorithm=\
-1/9*(x^4*log((x - log(x))*e^(e^8 + e^x))^2 - 9*x^5 - 4*x^2*log((x - log(x ))*e^(e^8 + e^x)) + 4)/x^4
Time = 3.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=x - \frac {\log {\left (\left (x - \log {\left (x \right )}\right ) e^{e^{x} + e^{8}} \right )}^{2}}{9} + \frac {4 \log {\left (\left (x - \log {\left (x \right )}\right ) e^{e^{x} + e^{8}} \right )}}{9 x^{2}} - \frac {4}{9 x^{4}} \]
integrate((((-2*x**5*exp(x)-8*x**2)*ln(x)+2*x**6*exp(x)+2*x**5-2*x**4+8*x* *3)*ln((x-ln(x))*exp(exp(x)+exp(4)**2))+(4*exp(x)*x**3+9*x**5+16)*ln(x)-4* exp(x)*x**4-9*x**6-4*x**3+4*x**2-16*x)/(9*x**5*ln(x)-9*x**6),x)
x - log((x - log(x))*exp(exp(x) + exp(8)))**2/9 + 4*log((x - log(x))*exp(e xp(x) + exp(8)))/(9*x**2) - 4/(9*x**4)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (28) = 56\).
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.52 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=-\frac {x^{4} \log \left (x - \log \left (x\right )\right )^{2} - 9 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{8} + 2 \, {\left (x^{4} e^{8} - 2 \, x^{2}\right )} e^{x} + 2 \, {\left (x^{4} e^{8} + x^{4} e^{x} - 2 \, x^{2}\right )} \log \left (x - \log \left (x\right )\right ) + 4}{9 \, x^{4}} \]
integrate((((-2*x^5*exp(x)-8*x^2)*log(x)+2*x^6*exp(x)+2*x^5-2*x^4+8*x^3)*l og((x-log(x))*exp(exp(x)+exp(4)^2))+(4*exp(x)*x^3+9*x^5+16)*log(x)-4*exp(x )*x^4-9*x^6-4*x^3+4*x^2-16*x)/(9*x^5*log(x)-9*x^6),x, algorithm=\
-1/9*(x^4*log(x - log(x))^2 - 9*x^5 + x^4*e^(2*x) - 4*x^2*e^8 + 2*(x^4*e^8 - 2*x^2)*e^x + 2*(x^4*e^8 + x^4*e^x - 2*x^2)*log(x - log(x)) + 4)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=-\frac {2 \, x^{4} e^{x} \log \left (x - \log \left (x\right )\right ) + x^{4} \log \left (x - \log \left (x\right )\right )^{2} + 2 \, x^{4} e^{8} \log \left (-x + \log \left (x\right )\right ) - 9 \, x^{5} + x^{4} e^{\left (2 \, x\right )} + 2 \, x^{4} e^{\left (x + 8\right )} - 4 \, x^{2} e^{8} - 4 \, x^{2} e^{x} - 4 \, x^{2} \log \left (x - \log \left (x\right )\right ) + 4}{9 \, x^{4}} \]
integrate((((-2*x^5*exp(x)-8*x^2)*log(x)+2*x^6*exp(x)+2*x^5-2*x^4+8*x^3)*l og((x-log(x))*exp(exp(x)+exp(4)^2))+(4*exp(x)*x^3+9*x^5+16)*log(x)-4*exp(x )*x^4-9*x^6-4*x^3+4*x^2-16*x)/(9*x^5*log(x)-9*x^6),x, algorithm=\
-1/9*(2*x^4*e^x*log(x - log(x)) + x^4*log(x - log(x))^2 + 2*x^4*e^8*log(-x + log(x)) - 9*x^5 + x^4*e^(2*x) + 2*x^4*e^(x + 8) - 4*x^2*e^8 - 4*x^2*e^x - 4*x^2*log(x - log(x)) + 4)/x^4
Time = 10.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-16 x+4 x^2-4 x^3-4 e^x x^4-9 x^6+\left (16+4 e^x x^3+9 x^5\right ) \log (x)+\left (8 x^3-2 x^4+2 x^5+2 e^x x^6+\left (-8 x^2-2 e^x x^5\right ) \log (x)\right ) \log \left (e^{e^8+e^x} (x-\log (x))\right )}{-9 x^6+9 x^5 \log (x)} \, dx=x+\frac {4\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^8}\,\left (x-\ln \left (x\right )\right )\right )}{9\,x^2}-\frac {{\ln \left ({\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^8}\,\left (x-\ln \left (x\right )\right )\right )}^2}{9}-\frac {4}{9\,x^4} \]
int(-(16*x + 4*x^4*exp(x) - log(exp(exp(8) + exp(x))*(x - log(x)))*(2*x^6* exp(x) + 8*x^3 - 2*x^4 + 2*x^5 - log(x)*(2*x^5*exp(x) + 8*x^2)) - 4*x^2 + 4*x^3 + 9*x^6 - log(x)*(4*x^3*exp(x) + 9*x^5 + 16))/(9*x^5*log(x) - 9*x^6) ,x)