Integrand size = 185, antiderivative size = 30 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx=(-2+x)^2 \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right ) \]
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx=(-2+x)^2 \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right ) \]
Integrate[((16 - 12*x^2 + 4*x^3 + E^3*(-16*x^2 - 16*x^3 + 28*x^4 - 8*x^5) + E^3*(-16*x - 16*x^2 + 28*x^3 - 8*x^4)*Log[x])*Log[Log[(x + Log[x])/E^(E^ 3*(2 + x + x^2))]^2] + (-4*x^2 + 2*x^3 + (-4*x + 2*x^2)*Log[x])*Log[(x + L og[x])/E^(E^3*(2 + x + x^2))]*Log[Log[(x + Log[x])/E^(E^3*(2 + x + x^2))]^ 2]^2)/((x^2 + x*Log[x])*Log[(x + Log[x])/E^(E^3*(2 + x + 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.
| \(\displaystyle \int \frac {\left (2 x^3-4 x^2+\left (2 x^2-4 x\right ) \log (x)\right ) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )+\left (4 x^3-12 x^2+e^3 \left (-8 x^4+28 x^3-16 x^2-16 x\right ) \log (x)+e^3 \left (-8 x^5+28 x^4-16 x^3-16 x^2\right )+16\right ) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )} \, dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {\left (2 x^3-4 x^2+\left (2 x^2-4 x\right ) \log (x)\right ) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )+\left (4 x^3-12 x^2+e^3 \left (-8 x^4+28 x^3-16 x^2-16 x\right ) \log (x)+e^3 \left (-8 x^5+28 x^4-16 x^3-16 x^2\right )+16\right ) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{x (x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 (x-2) \log ^2\left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )-\frac {4 (x-2)^2 \left (2 e^3 x^3+e^3 x^2+2 e^3 x^2 \log (x)-x+e^3 x \log (x)-1\right ) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{x (x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \int \frac {\log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{x (x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-16 e^3 \int \frac {x \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-12 \int \frac {x \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-16 e^3 \int \frac {x^2 \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx+4 \int \frac {x^2 \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-16 e^3 \int \frac {\log (x) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-16 e^3 \int \frac {x \log (x) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx+28 e^3 \int \frac {x^2 \log (x) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-4 \int \log ^2\left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )dx+2 \int x \log ^2\left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )dx-8 e^3 \int \frac {x^4 \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx+28 e^3 \int \frac {x^3 \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx-8 e^3 \int \frac {x^3 \log (x) \log \left (\log ^2\left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )\right )}{(x+\log (x)) \log \left (e^{-e^3 \left (x^2+x+2\right )} (x+\log (x))\right )}dx\) |
Int[((16 - 12*x^2 + 4*x^3 + E^3*(-16*x^2 - 16*x^3 + 28*x^4 - 8*x^5) + E^3* (-16*x - 16*x^2 + 28*x^3 - 8*x^4)*Log[x])*Log[Log[(x + Log[x])/E^(E^3*(2 + x + x^2))]^2] + (-4*x^2 + 2*x^3 + (-4*x + 2*x^2)*Log[x])*Log[(x + Log[x]) /E^(E^3*(2 + x + x^2))]*Log[Log[(x + Log[x])/E^(E^3*(2 + x + x^2))]^2]^2)/ ((x^2 + x*Log[x])*Log[(x + Log[x])/E^(E^3*(2 + x + x^2))]),x]
3.25.23.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(29)=58\).
Time = 226.96 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67
| method | result | size |
| parallelrisch | \(x^{2} {\ln \left (\ln \left (\left (x +\ln \left (x \right )\right ) {\mathrm e}^{-\left (x^{2}+x +2\right ) {\mathrm e}^{3}}\right )^{2}\right )}^{2}-4 {\ln \left (\ln \left (\left (x +\ln \left (x \right )\right ) {\mathrm e}^{-\left (x^{2}+x +2\right ) {\mathrm e}^{3}}\right )^{2}\right )}^{2} x +4 {\ln \left (\ln \left (\left (x +\ln \left (x \right )\right ) {\mathrm e}^{-\left (x^{2}+x +2\right ) {\mathrm e}^{3}}\right )^{2}\right )}^{2}\) | \(80\) |
int((((2*x^2-4*x)*ln(x)+2*x^3-4*x^2)*ln((x+ln(x))/exp((x^2+x+2)*exp(3)))*l n(ln((x+ln(x))/exp((x^2+x+2)*exp(3)))^2)^2+((-8*x^4+28*x^3-16*x^2-16*x)*ex p(3)*ln(x)+(-8*x^5+28*x^4-16*x^3-16*x^2)*exp(3)+4*x^3-12*x^2+16)*ln(ln((x+ ln(x))/exp((x^2+x+2)*exp(3)))^2))/(x*ln(x)+x^2)/ln((x+ln(x))/exp((x^2+x+2) *exp(3))),x,method=_RETURNVERBOSE)
x^2*ln(ln((x+ln(x))/exp((x^2+x+2)*exp(3)))^2)^2-4*ln(ln((x+ln(x))/exp((x^2 +x+2)*exp(3)))^2)^2*x+4*ln(ln((x+ln(x))/exp((x^2+x+2)*exp(3)))^2)^2
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx={\left (x^{2} - 4 \, x + 4\right )} \log \left (\log \left (x e^{\left (-{\left (x^{2} + x + 2\right )} e^{3}\right )} + e^{\left (-{\left (x^{2} + x + 2\right )} e^{3}\right )} \log \left (x\right )\right )^{2}\right )^{2} \]
integrate((((2*x^2-4*x)*log(x)+2*x^3-4*x^2)*log((x+log(x))/exp((x^2+x+2)*e xp(3)))*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2)^2+((-8*x^4+28*x^3-16* x^2-16*x)*exp(3)*log(x)+(-8*x^5+28*x^4-16*x^3-16*x^2)*exp(3)+4*x^3-12*x^2+ 16)*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2))/(x*log(x)+x^2)/log((x+lo g(x))/exp((x^2+x+2)*exp(3))),x, algorithm=\
Exception generated. \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx=\text {Exception raised: TypeError} \]
integrate((((2*x**2-4*x)*ln(x)+2*x**3-4*x**2)*ln((x+ln(x))/exp((x**2+x+2)* exp(3)))*ln(ln((x+ln(x))/exp((x**2+x+2)*exp(3)))**2)**2+((-8*x**4+28*x**3- 16*x**2-16*x)*exp(3)*ln(x)+(-8*x**5+28*x**4-16*x**3-16*x**2)*exp(3)+4*x**3 -12*x**2+16)*ln(ln((x+ln(x))/exp((x**2+x+2)*exp(3)))**2))/(x*ln(x)+x**2)/l n((x+ln(x))/exp((x**2+x+2)*exp(3))),x)
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx=4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (x^{2} e^{3} + x e^{3} + 2 \, e^{3} - \log \left (x + \log \left (x\right )\right )\right )^{2} \]
integrate((((2*x^2-4*x)*log(x)+2*x^3-4*x^2)*log((x+log(x))/exp((x^2+x+2)*e xp(3)))*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2)^2+((-8*x^4+28*x^3-16* x^2-16*x)*exp(3)*log(x)+(-8*x^5+28*x^4-16*x^3-16*x^2)*exp(3)+4*x^3-12*x^2+ 16)*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2))/(x*log(x)+x^2)/log((x+lo g(x))/exp((x^2+x+2)*exp(3))),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (28) = 56\).
Time = 179.55 (sec) , antiderivative size = 223, normalized size of antiderivative = 7.43 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx=x^{2} \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} - 4 \, x \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} + 4 \, \log \left (x^{4} e^{6} + 2 \, x^{3} e^{6} - 2 \, x^{2} e^{3} \log \left (x + \log \left (x\right )\right ) + 5 \, x^{2} e^{6} - 2 \, x e^{3} \log \left (x + \log \left (x\right )\right ) + 4 \, x e^{6} - 4 \, e^{3} \log \left (x + \log \left (x\right )\right ) + \log \left (x + \log \left (x\right )\right )^{2} + 4 \, e^{6}\right )^{2} \]
integrate((((2*x^2-4*x)*log(x)+2*x^3-4*x^2)*log((x+log(x))/exp((x^2+x+2)*e xp(3)))*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2)^2+((-8*x^4+28*x^3-16* x^2-16*x)*exp(3)*log(x)+(-8*x^5+28*x^4-16*x^3-16*x^2)*exp(3)+4*x^3-12*x^2+ 16)*log(log((x+log(x))/exp((x^2+x+2)*exp(3)))^2))/(x*log(x)+x^2)/log((x+lo g(x))/exp((x^2+x+2)*exp(3))),x, algorithm=\
x^2*log(x^4*e^6 + 2*x^3*e^6 - 2*x^2*e^3*log(x + log(x)) + 5*x^2*e^6 - 2*x* e^3*log(x + log(x)) + 4*x*e^6 - 4*e^3*log(x + log(x)) + log(x + log(x))^2 + 4*e^6)^2 - 4*x*log(x^4*e^6 + 2*x^3*e^6 - 2*x^2*e^3*log(x + log(x)) + 5*x ^2*e^6 - 2*x*e^3*log(x + log(x)) + 4*x*e^6 - 4*e^3*log(x + log(x)) + log(x + log(x))^2 + 4*e^6)^2 + 4*log(x^4*e^6 + 2*x^3*e^6 - 2*x^2*e^3*log(x + lo g(x)) + 5*x^2*e^6 - 2*x*e^3*log(x + log(x)) + 4*x*e^6 - 4*e^3*log(x + log( x)) + log(x + log(x))^2 + 4*e^6)^2
Time = 11.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {\left (16-12 x^2+4 x^3+e^3 \left (-16 x^2-16 x^3+28 x^4-8 x^5\right )+e^3 \left (-16 x-16 x^2+28 x^3-8 x^4\right ) \log (x)\right ) \log \left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )+\left (-4 x^2+2 x^3+\left (-4 x+2 x^2\right ) \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right ) \log ^2\left (\log ^2\left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )\right )}{\left (x^2+x \log (x)\right ) \log \left (e^{-e^3 \left (2+x+x^2\right )} (x+\log (x))\right )} \, dx={\ln \left ({\ln \left ({\mathrm {e}}^{-x^2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^3}\,\left (x+\ln \left (x\right )\right )\right )}^2\right )}^2\,{\left (x-2\right )}^2 \]
int(-(log(log(exp(-exp(3)*(x + x^2 + 2))*(x + log(x)))^2)*(12*x^2 - 4*x^3 + exp(3)*(16*x^2 + 16*x^3 - 28*x^4 + 8*x^5) + exp(3)*log(x)*(16*x + 16*x^2 - 28*x^3 + 8*x^4) - 16) + log(log(exp(-exp(3)*(x + x^2 + 2))*(x + log(x)) )^2)^2*log(exp(-exp(3)*(x + x^2 + 2))*(x + log(x)))*(log(x)*(4*x - 2*x^2) + 4*x^2 - 2*x^3))/(log(exp(-exp(3)*(x + x^2 + 2))*(x + log(x)))*(x*log(x) + x^2)),x)