Integrand size = 208, antiderivative size = 25 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{4 \left (\frac {x}{2}+\frac {x \log (4)}{(1+x)^2 (x+\log (x))}\right )} \]
\[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx \]
Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)* Log[x])/(x + 2*x^2 + x^3 + (1 + 2*x + x^2)*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 + (4 - 4*x)*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)*Log[x]^2),x]
Integrate[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)* Log[x])/(x + 2*x^2 + x^3 + (1 + 2*x + x^2)*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 + (4 - 4*x)*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)*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.
| \(\displaystyle \int \frac {\left (2 x^5+6 x^4+6 x^3+2 x^2+\left (-8 x^2-4 x-4\right ) \log (4)+\left (2 x^3+6 x^2+6 x+2\right ) \log ^2(x)+\left (4 x^4+12 x^3+12 x^2+4 x+(4-4 x) \log (4)\right ) \log (x)\right ) \exp \left (\frac {2 x^4+4 x^3+2 x^2+\left (2 x^3+4 x^2+2 x\right ) \log (x)+4 x \log (4)}{x^3+2 x^2+\left (x^2+2 x+1\right ) \log (x)+x}\right )}{x^5+3 x^4+3 x^3+x^2+\left (x^3+3 x^2+3 x+1\right ) \log ^2(x)+\left (2 x^4+6 x^3+6 x^2+2 x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+1} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}} \left (x^5+3 x^4+3 x^3+x^2 (1-4 \log (4))+2 \left (x^4+3 x^3+3 x^2+x-x \log (4)+\log (4)\right ) \log (x)+(x+1)^3 \log ^2(x)-2 x \log (4)-2 \log (4)\right )}{(x+1)^3 (x+\log (x))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+1} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}-\frac {(x-1) \log (4) 2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+2} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(x+1)^3 (x+\log (x))}-\frac {\log (4) 2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+2} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(x+1) (x+\log (x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int 2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+1} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}dx-\log (4) \int \frac {2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+2} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(x+1) (x+\log (x))^2}dx+\log (4) \int \frac {2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+3} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(x+1)^3 (x+\log (x))}dx-\log (4) \int \frac {2^{\frac {8 x}{(x+1)^2 (x+\log (x))}+2} e^{\frac {2 x^2}{x+\log (x)}} x^{\frac {2 x}{x+\log (x)}}}{(x+1)^2 (x+\log (x))}dx\) |
Int[(E^((2*x^2 + 4*x^3 + 2*x^4 + 4*x*Log[4] + (2*x + 4*x^2 + 2*x^3)*Log[x] )/(x + 2*x^2 + x^3 + (1 + 2*x + x^2)*Log[x]))*(2*x^2 + 6*x^3 + 6*x^4 + 2*x ^5 + (-4 - 4*x - 8*x^2)*Log[4] + (4*x + 12*x^2 + 12*x^3 + 4*x^4 + (4 - 4*x )*Log[4])*Log[x] + (2 + 6*x + 6*x^2 + 2*x^3)*Log[x]^2))/(x^2 + 3*x^3 + 3*x ^4 + x^5 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[x] + (1 + 3*x + 3*x^2 + x^3)* Log[x]^2),x]
3.22.13.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 21.91 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72
| method | result | size |
| risch | \({\mathrm e}^{\frac {2 x \left (x^{2} \ln \left (x \right )+x^{3}+2 x \ln \left (x \right )+2 x^{2}+\ln \left (x \right )+4 \ln \left (2\right )+x \right )}{\left (1+x \right )^{2} \left (x +\ln \left (x \right )\right )}}\) | \(43\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (2 x^{3}+4 x^{2}+2 x \right ) \ln \left (x \right )+8 x \ln \left (2\right )+2 x^{4}+4 x^{3}+2 x^{2}}{x^{2} \ln \left (x \right )+x^{3}+2 x \ln \left (x \right )+2 x^{2}+\ln \left (x \right )+x}}\) | \(66\) |
int(((2*x^3+6*x^2+6*x+2)*ln(x)^2+(2*(4-4*x)*ln(2)+4*x^4+12*x^3+12*x^2+4*x) *ln(x)+2*(-8*x^2-4*x-4)*ln(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x^3+4*x^2+2 *x)*ln(x)+8*x*ln(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*ln(x)+x^3+2*x^2+x))/(( x^3+3*x^2+3*x+1)*ln(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*ln(x)+x^5+3*x^4+3*x^3+x^2 ),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (\frac {2 \, {\left (x^{4} + 2 \, x^{3} + x^{2} + 4 \, x \log \left (2\right ) + {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (x\right )\right )}}{x^{3} + 2 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right ) + x}\right )} \]
integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(4-4*x)*log(2)+4*x^4+12*x^3+12* x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*log(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x ^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+x^3 +2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+ 3*x^4+3*x^3+x^2),x, algorithm=\
e^(2*(x^4 + 2*x^3 + x^2 + 4*x*log(2) + (x^3 + 2*x^2 + x)*log(x))/(x^3 + 2* x^2 + (x^2 + 2*x + 1)*log(x) + x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\frac {2 x^{4} + 4 x^{3} + 2 x^{2} + 8 x \log {\left (2 \right )} + \left (2 x^{3} + 4 x^{2} + 2 x\right ) \log {\left (x \right )}}{x^{3} + 2 x^{2} + x + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}}} \]
integrate(((2*x**3+6*x**2+6*x+2)*ln(x)**2+(2*(4-4*x)*ln(2)+4*x**4+12*x**3+ 12*x**2+4*x)*ln(x)+2*(-8*x**2-4*x-4)*ln(2)+2*x**5+6*x**4+6*x**3+2*x**2)*ex p(((2*x**3+4*x**2+2*x)*ln(x)+8*x*ln(2)+2*x**4+4*x**3+2*x**2)/((x**2+2*x+1) *ln(x)+x**3+2*x**2+x))/((x**3+3*x**2+3*x+1)*ln(x)**2+(2*x**4+6*x**3+6*x**2 +2*x)*ln(x)+x**5+3*x**4+3*x**3+x**2),x)
exp((2*x**4 + 4*x**3 + 2*x**2 + 8*x*log(2) + (2*x**3 + 4*x**2 + 2*x)*log(x ))/(x**3 + 2*x**2 + x + (x**2 + 2*x + 1)*log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (21) = 42\).
Time = 0.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (2 \, x + \frac {8 \, \log \left (2\right ) \log \left (x\right )}{{\left (x + 1\right )} \log \left (x\right )^{2} - 2 \, {\left (x + 1\right )} \log \left (x\right ) + x + 1} - \frac {8 \, \log \left (2\right ) \log \left (x\right )}{{\left (x - 2\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} - {\left (2 \, x - 1\right )} \log \left (x\right ) + x} + \frac {8 \, \log \left (2\right )}{x^{2} - {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right ) + 2 \, x + 1}\right )} \]
integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(4-4*x)*log(2)+4*x^4+12*x^3+12* x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*log(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x ^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+x^3 +2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+ 3*x^4+3*x^3+x^2),x, algorithm=\
e^(2*x + 8*log(2)*log(x)/((x + 1)*log(x)^2 - 2*(x + 1)*log(x) + x + 1) - 8 *log(2)*log(x)/((x - 2)*log(x)^2 + log(x)^3 - (2*x - 1)*log(x) + x) + 8*lo g(2)/(x^2 - (x^2 + 2*x + 1)*log(x) + 2*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (21) = 42\).
Time = 0.63 (sec) , antiderivative size = 216, normalized size of antiderivative = 8.64 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx=e^{\left (\frac {2 \, x^{4}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x^{3} \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {4 \, x^{3}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {4 \, x^{2} \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x^{2}}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {8 \, x \log \left (2\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )} + \frac {2 \, x \log \left (x\right )}{x^{3} + x^{2} \log \left (x\right ) + 2 \, x^{2} + 2 \, x \log \left (x\right ) + x + \log \left (x\right )}\right )} \]
integrate(((2*x^3+6*x^2+6*x+2)*log(x)^2+(2*(4-4*x)*log(2)+4*x^4+12*x^3+12* x^2+4*x)*log(x)+2*(-8*x^2-4*x-4)*log(2)+2*x^5+6*x^4+6*x^3+2*x^2)*exp(((2*x ^3+4*x^2+2*x)*log(x)+8*x*log(2)+2*x^4+4*x^3+2*x^2)/((x^2+2*x+1)*log(x)+x^3 +2*x^2+x))/((x^3+3*x^2+3*x+1)*log(x)^2+(2*x^4+6*x^3+6*x^2+2*x)*log(x)+x^5+ 3*x^4+3*x^3+x^2),x, algorithm=\
e^(2*x^4/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 2*x^3*log( x)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 4*x^3/(x^3 + x^2 *log(x) + 2*x^2 + 2*x*log(x) + x + log(x)) + 4*x^2*log(x)/(x^3 + x^2*log(x ) + 2*x^2 + 2*x*log(x) + x + log(x)) + 2*x^2/(x^3 + x^2*log(x) + 2*x^2 + 2 *x*log(x) + x + log(x)) + 8*x*log(2)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x ) + x + log(x)) + 2*x*log(x)/(x^3 + x^2*log(x) + 2*x^2 + 2*x*log(x) + x + log(x)))
Time = 13.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.36 \[ \int \frac {e^{\frac {2 x^2+4 x^3+2 x^4+4 x \log (4)+\left (2 x+4 x^2+2 x^3\right ) \log (x)}{x+2 x^2+x^3+\left (1+2 x+x^2\right ) \log (x)}} \left (2 x^2+6 x^3+6 x^4+2 x^5+\left (-4-4 x-8 x^2\right ) \log (4)+\left (4 x+12 x^2+12 x^3+4 x^4+(4-4 x) \log (4)\right ) \log (x)+\left (2+6 x+6 x^2+2 x^3\right ) \log ^2(x)\right )}{x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (x)+\left (1+3 x+3 x^2+x^3\right ) \log ^2(x)} \, dx={256}^{\frac {x}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,x^{\frac {2\,x}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {2\,x^2}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {2\,x^4}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}}\,{\mathrm {e}}^{\frac {4\,x^3}{x+\ln \left (x\right )+x^2\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+2\,x^2+x^3}} \]
int((exp((8*x*log(2) + 2*x^2 + 4*x^3 + 2*x^4 + log(x)*(2*x + 4*x^2 + 2*x^3 ))/(x + log(x)*(2*x + x^2 + 1) + 2*x^2 + x^3))*(log(x)*(4*x - 2*log(2)*(4* x - 4) + 12*x^2 + 12*x^3 + 4*x^4) - 2*log(2)*(4*x + 8*x^2 + 4) + log(x)^2* (6*x + 6*x^2 + 2*x^3 + 2) + 2*x^2 + 6*x^3 + 6*x^4 + 2*x^5))/(log(x)*(2*x + 6*x^2 + 6*x^3 + 2*x^4) + x^2 + 3*x^3 + 3*x^4 + x^5 + log(x)^2*(3*x + 3*x^ 2 + x^3 + 1)),x)