Integrand size = 266, antiderivative size = 28 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=-e^x+\left (\frac {2 \log (x)}{1+x}+\log \left (\log \left (\frac {x}{2}+\log (2)\right )\right )\right )^2 \] Output:
(ln(ln(ln(2)+1/2*x))+2*ln(x)/(1+x))^2-exp(x)
\[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx \] Input:
Integrate[((4*x + 8*x^2 + 4*x^3)*Log[x] + (E^x*(-x^2 - 3*x^3 - 3*x^4 - x^5 + (-2*x - 6*x^2 - 6*x^3 - 2*x^4)*Log[2]) + (8*x + 8*x^2 + (16 + 16*x)*Log [2])*Log[x] + (-8*x^2 - 16*x*Log[2])*Log[x]^2)*Log[(x + 2*Log[2])/2] + (2* x + 6*x^2 + 6*x^3 + 2*x^4 + (4*x + 8*x^2 + 4*x^3 + (8 + 16*x + 8*x^2)*Log[ 2] + (-4*x^2 - 4*x^3 + (-8*x - 8*x^2)*Log[2])*Log[x])*Log[(x + 2*Log[2])/2 ])*Log[Log[(x + 2*Log[2])/2]])/((x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[2])*Log[(x + 2*Log[2])/2]),x]
Output:
Integrate[((4*x + 8*x^2 + 4*x^3)*Log[x] + (E^x*(-x^2 - 3*x^3 - 3*x^4 - x^5 + (-2*x - 6*x^2 - 6*x^3 - 2*x^4)*Log[2]) + (8*x + 8*x^2 + (16 + 16*x)*Log [2])*Log[x] + (-8*x^2 - 16*x*Log[2])*Log[x]^2)*Log[(x + 2*Log[2])/2] + (2* x + 6*x^2 + 6*x^3 + 2*x^4 + (4*x + 8*x^2 + 4*x^3 + (8 + 16*x + 8*x^2)*Log[ 2] + (-4*x^2 - 4*x^3 + (-8*x - 8*x^2)*Log[2])*Log[x])*Log[(x + 2*Log[2])/2 ])*Log[Log[(x + 2*Log[2])/2]])/((x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[2])*Log[(x + 2*Log[2])/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.
| \(\displaystyle \int \frac {\left (4 x^3+8 x^2+4 x\right ) \log (x)+\left (2 x^4+6 x^3+6 x^2+\left (4 x^3+8 x^2+\left (8 x^2+16 x+8\right ) \log (2)+\left (-4 x^3-4 x^2+\left (-8 x^2-8 x\right ) \log (2)\right ) \log (x)+4 x\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+2 x\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )+\left (\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)+\left (8 x^2+8 x+(16 x+16) \log (2)\right ) \log (x)+e^x \left (-x^5-3 x^4-3 x^3-x^2+\left (-2 x^4-6 x^3-6 x^2-2 x\right ) \log (2)\right )\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )}{\left (x^5+3 x^4+3 x^3+x^2+\left (2 x^4+6 x^3+6 x^2+2 x\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (4 x^3+8 x^2+4 x\right ) \log (x)+\left (2 x^4+6 x^3+6 x^2+\left (4 x^3+8 x^2+\left (8 x^2+16 x+8\right ) \log (2)+\left (-4 x^3-4 x^2+\left (-8 x^2-8 x\right ) \log (2)\right ) \log (x)+4 x\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+2 x\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )+\left (\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)+\left (8 x^2+8 x+(16 x+16) \log (2)\right ) \log (x)+e^x \left (-x^5-3 x^4-3 x^3-x^2+\left (-2 x^4-6 x^3-6 x^2-2 x\right ) \log (2)\right )\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )}{x \left (x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (x^4+3 x^3+3 x^2+x-2 (x+1) (x+\log (4)) (-x+x \log (x)-1) \log \left (\frac {x}{2}+\log (2)\right )\right ) \log \left (\log \left (\frac {x}{2}+\log (2)\right )\right )-(x+\log (4)) \left (e^x x (x+1)^3+8 x \log ^2(x)-8 (x+1) \log (x)\right ) \log \left (\frac {x}{2}+\log (2)\right )+4 x (x+1)^2 \log (x)}{x \left (x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)\right ) \log \left (\frac {x}{2}+\log (2)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^x (x+1)^2 (-x-\log (4))}{x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)}+\frac {4 (x+1) \log (x)}{\left (x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)\right ) \log \left (\frac {x}{2}+\log (2)\right )}+\frac {8 (x+\log (4)) \log (x)}{x \left (x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)\right )}+\frac {8 (-x-\log (4)) \log ^2(x)}{x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)}+\frac {2 (x+1) \left (x^3+2 x^2-2 x^2 \log (x) \log \left (\frac {x}{2}+\log (2)\right )+2 x^2 \log \left (\frac {x}{2}+\log (2)\right )+x-2 x \log (4) \log (x) \log \left (\frac {x}{2}+\log (2)\right )+2 x (1+\log (4)) \log \left (\frac {x}{2}+\log (2)\right )+2 \log (4) \log \left (\frac {x}{2}+\log (2)\right )\right ) \log \left (\log \left (\frac {x}{2}+\log (2)\right )\right )}{x \left (x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)\right ) \log \left (\frac {x}{2}+\log (2)\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {e^x (x+1)^2 (-x-\log (4))}{x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)}+\frac {4 (x+1) \log (x)}{\left (x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)\right ) \log \left (\frac {x}{2}+\log (2)\right )}+\frac {8 (x+\log (4)) \log (x)}{x \left (x^3+x^2 (2+\log (4))+x (1+\log (16))+\log (4)\right )}+\frac {8 (-x-\log (4)) \log ^2(x)}{x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)}+\frac {2 (x+1) \left (x^3+2 x^2-2 x^2 \log (x) \log \left (\frac {x}{2}+\log (2)\right )+2 x^2 \log \left (\frac {x}{2}+\log (2)\right )+x-2 x \log (4) \log (x) \log \left (\frac {x}{2}+\log (2)\right )+2 x (1+\log (4)) \log \left (\frac {x}{2}+\log (2)\right )+2 \log (4) \log \left (\frac {x}{2}+\log (2)\right )\right ) \log \left (\log \left (\frac {x}{2}+\log (2)\right )\right )}{x \left (x^4+x^3 (3+\log (4))+3 x^2 (1+\log (4))+x (1+\log (64))+\log (4)\right ) \log \left (\frac {x}{2}+\log (2)\right )}\right )dx\) |
Input:
Int[((4*x + 8*x^2 + 4*x^3)*Log[x] + (E^x*(-x^2 - 3*x^3 - 3*x^4 - x^5 + (-2 *x - 6*x^2 - 6*x^3 - 2*x^4)*Log[2]) + (8*x + 8*x^2 + (16 + 16*x)*Log[2])*L og[x] + (-8*x^2 - 16*x*Log[2])*Log[x]^2)*Log[(x + 2*Log[2])/2] + (2*x + 6* x^2 + 6*x^3 + 2*x^4 + (4*x + 8*x^2 + 4*x^3 + (8 + 16*x + 8*x^2)*Log[2] + ( -4*x^2 - 4*x^3 + (-8*x - 8*x^2)*Log[2])*Log[x])*Log[(x + 2*Log[2])/2])*Log [Log[(x + 2*Log[2])/2]])/((x^2 + 3*x^3 + 3*x^4 + x^5 + (2*x + 6*x^2 + 6*x^ 3 + 2*x^4)*Log[2])*Log[(x + 2*Log[2])/2]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
Time = 0.43 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00
\[\ln \left (\ln \left (\ln \left (2\right )+\frac {x}{2}\right )\right )^{2}+\frac {4 \ln \left (x \right ) \ln \left (\ln \left (\ln \left (2\right )+\frac {x}{2}\right )\right )}{1+x}-\frac {{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x -4 \ln \left (x \right )^{2}+{\mathrm e}^{x}}{\left (1+x \right )^{2}}\]
Input:
int((((((-8*x^2-8*x)*ln(2)-4*x^3-4*x^2)*ln(x)+(8*x^2+16*x+8)*ln(2)+4*x^3+8 *x^2+4*x)*ln(ln(2)+1/2*x)+2*x^4+6*x^3+6*x^2+2*x)*ln(ln(ln(2)+1/2*x))+((-16 *x*ln(2)-8*x^2)*ln(x)^2+((16*x+16)*ln(2)+8*x^2+8*x)*ln(x)+((-2*x^4-6*x^3-6 *x^2-2*x)*ln(2)-x^5-3*x^4-3*x^3-x^2)*exp(x))*ln(ln(2)+1/2*x)+(4*x^3+8*x^2+ 4*x)*ln(x))/((2*x^4+6*x^3+6*x^2+2*x)*ln(2)+x^5+3*x^4+3*x^3+x^2)/ln(ln(2)+1 /2*x),x)
Output:
ln(ln(ln(2)+1/2*x))^2+4/(1+x)*ln(x)*ln(ln(ln(2)+1/2*x))-(exp(x)*x^2+2*exp( x)*x-4*ln(x)^2+exp(x))/(1+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.29 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\frac {4 \, {\left (x + 1\right )} \log \left (x\right ) \log \left (\log \left (\frac {1}{2} \, x + \log \left (2\right )\right )\right ) + {\left (x^{2} + 2 \, x + 1\right )} \log \left (\log \left (\frac {1}{2} \, x + \log \left (2\right )\right )\right )^{2} - {\left (x^{2} + 2 \, x + 1\right )} e^{x} + 4 \, \log \left (x\right )^{2}}{x^{2} + 2 \, x + 1} \] Input:
integrate((((((-8*x^2-8*x)*log(2)-4*x^3-4*x^2)*log(x)+(8*x^2+16*x+8)*log(2 )+4*x^3+8*x^2+4*x)*log(log(2)+1/2*x)+2*x^4+6*x^3+6*x^2+2*x)*log(log(log(2) +1/2*x))+((-16*x*log(2)-8*x^2)*log(x)^2+((16*x+16)*log(2)+8*x^2+8*x)*log(x )+((-2*x^4-6*x^3-6*x^2-2*x)*log(2)-x^5-3*x^4-3*x^3-x^2)*exp(x))*log(log(2) +1/2*x)+(4*x^3+8*x^2+4*x)*log(x))/((2*x^4+6*x^3+6*x^2+2*x)*log(2)+x^5+3*x^ 4+3*x^3+x^2)/log(log(2)+1/2*x),x, algorithm="fricas")
Output:
(4*(x + 1)*log(x)*log(log(1/2*x + log(2))) + (x^2 + 2*x + 1)*log(log(1/2*x + log(2)))^2 - (x^2 + 2*x + 1)*e^x + 4*log(x)^2)/(x^2 + 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 1.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=- e^{x} + \log {\left (\log {\left (\frac {x}{2} + \log {\left (2 \right )} \right )} \right )}^{2} + \frac {4 \log {\left (x \right )}^{2}}{x^{2} + 2 x + 1} + \frac {4 \log {\left (x \right )} \log {\left (\log {\left (\frac {x}{2} + \log {\left (2 \right )} \right )} \right )}}{x + 1} \] Input:
integrate((((((-8*x**2-8*x)*ln(2)-4*x**3-4*x**2)*ln(x)+(8*x**2+16*x+8)*ln( 2)+4*x**3+8*x**2+4*x)*ln(ln(2)+1/2*x)+2*x**4+6*x**3+6*x**2+2*x)*ln(ln(ln(2 )+1/2*x))+((-16*x*ln(2)-8*x**2)*ln(x)**2+((16*x+16)*ln(2)+8*x**2+8*x)*ln(x )+((-2*x**4-6*x**3-6*x**2-2*x)*ln(2)-x**5-3*x**4-3*x**3-x**2)*exp(x))*ln(l n(2)+1/2*x)+(4*x**3+8*x**2+4*x)*ln(x))/((2*x**4+6*x**3+6*x**2+2*x)*ln(2)+x **5+3*x**4+3*x**3+x**2)/ln(ln(2)+1/2*x),x)
Output:
-exp(x) + log(log(x/2 + log(2)))**2 + 4*log(x)**2/(x**2 + 2*x + 1) + 4*log (x)*log(log(x/2 + log(2)))/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\frac {4 \, {\left (x + 1\right )} \log \left (x\right ) \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right ) + {\left (x^{2} + 2 \, x + 1\right )} \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right )^{2} - {\left (x^{2} + 2 \, x + 1\right )} e^{x} + 4 \, \log \left (x\right )^{2}}{x^{2} + 2 \, x + 1} \] Input:
integrate((((((-8*x^2-8*x)*log(2)-4*x^3-4*x^2)*log(x)+(8*x^2+16*x+8)*log(2 )+4*x^3+8*x^2+4*x)*log(log(2)+1/2*x)+2*x^4+6*x^3+6*x^2+2*x)*log(log(log(2) +1/2*x))+((-16*x*log(2)-8*x^2)*log(x)^2+((16*x+16)*log(2)+8*x^2+8*x)*log(x )+((-2*x^4-6*x^3-6*x^2-2*x)*log(2)-x^5-3*x^4-3*x^3-x^2)*exp(x))*log(log(2) +1/2*x)+(4*x^3+8*x^2+4*x)*log(x))/((2*x^4+6*x^3+6*x^2+2*x)*log(2)+x^5+3*x^ 4+3*x^3+x^2)/log(log(2)+1/2*x),x, algorithm="maxima")
Output:
(4*(x + 1)*log(x)*log(-log(2) + log(x + 2*log(2))) + (x^2 + 2*x + 1)*log(- log(2) + log(x + 2*log(2)))^2 - (x^2 + 2*x + 1)*e^x + 4*log(x)^2)/(x^2 + 2 *x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.32 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\frac {x^{2} \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right )^{2} - x^{2} e^{x} + 4 \, x \log \left (x\right ) \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right ) + 2 \, x \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right )^{2} - 2 \, x e^{x} + 4 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right ) + \log \left (-\log \left (2\right ) + \log \left (x + 2 \, \log \left (2\right )\right )\right )^{2} - e^{x}}{x^{2} + 2 \, x + 1} \] Input:
integrate((((((-8*x^2-8*x)*log(2)-4*x^3-4*x^2)*log(x)+(8*x^2+16*x+8)*log(2 )+4*x^3+8*x^2+4*x)*log(log(2)+1/2*x)+2*x^4+6*x^3+6*x^2+2*x)*log(log(log(2) +1/2*x))+((-16*x*log(2)-8*x^2)*log(x)^2+((16*x+16)*log(2)+8*x^2+8*x)*log(x )+((-2*x^4-6*x^3-6*x^2-2*x)*log(2)-x^5-3*x^4-3*x^3-x^2)*exp(x))*log(log(2) +1/2*x)+(4*x^3+8*x^2+4*x)*log(x))/((2*x^4+6*x^3+6*x^2+2*x)*log(2)+x^5+3*x^ 4+3*x^3+x^2)/log(log(2)+1/2*x),x, algorithm="giac")
Output:
(x^2*log(-log(2) + log(x + 2*log(2)))^2 - x^2*e^x + 4*x*log(x)*log(-log(2) + log(x + 2*log(2))) + 2*x*log(-log(2) + log(x + 2*log(2)))^2 - 2*x*e^x + 4*log(x)^2 + 4*log(x)*log(-log(2) + log(x + 2*log(2))) + log(-log(2) + lo g(x + 2*log(2)))^2 - e^x)/(x^2 + 2*x + 1)
Timed out. \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\int \frac {\ln \left (\ln \left (\frac {x}{2}+\ln \left (2\right )\right )\right )\,\left (2\,x+\ln \left (\frac {x}{2}+\ln \left (2\right )\right )\,\left (4\,x+\ln \left (2\right )\,\left (8\,x^2+16\,x+8\right )-\ln \left (x\right )\,\left (\ln \left (2\right )\,\left (8\,x^2+8\,x\right )+4\,x^2+4\,x^3\right )+8\,x^2+4\,x^3\right )+6\,x^2+6\,x^3+2\,x^4\right )-\ln \left (\frac {x}{2}+\ln \left (2\right )\right )\,\left (\left (8\,x^2+16\,\ln \left (2\right )\,x\right )\,{\ln \left (x\right )}^2+\left (-8\,x-\ln \left (2\right )\,\left (16\,x+16\right )-8\,x^2\right )\,\ln \left (x\right )+{\mathrm {e}}^x\,\left (\ln \left (2\right )\,\left (2\,x^4+6\,x^3+6\,x^2+2\,x\right )+x^2+3\,x^3+3\,x^4+x^5\right )\right )+\ln \left (x\right )\,\left (4\,x^3+8\,x^2+4\,x\right )}{\ln \left (\frac {x}{2}+\ln \left (2\right )\right )\,\left (\ln \left (2\right )\,\left (2\,x^4+6\,x^3+6\,x^2+2\,x\right )+x^2+3\,x^3+3\,x^4+x^5\right )} \,d x \] Input:
int((log(log(x/2 + log(2)))*(2*x + log(x/2 + log(2))*(4*x + log(2)*(16*x + 8*x^2 + 8) - log(x)*(log(2)*(8*x + 8*x^2) + 4*x^2 + 4*x^3) + 8*x^2 + 4*x^ 3) + 6*x^2 + 6*x^3 + 2*x^4) - log(x/2 + log(2))*(exp(x)*(log(2)*(2*x + 6*x ^2 + 6*x^3 + 2*x^4) + x^2 + 3*x^3 + 3*x^4 + x^5) + log(x)^2*(16*x*log(2) + 8*x^2) - log(x)*(8*x + log(2)*(16*x + 16) + 8*x^2)) + log(x)*(4*x + 8*x^2 + 4*x^3))/(log(x/2 + log(2))*(log(2)*(2*x + 6*x^2 + 6*x^3 + 2*x^4) + x^2 + 3*x^3 + 3*x^4 + x^5)),x)
Output:
int((log(log(x/2 + log(2)))*(2*x + log(x/2 + log(2))*(4*x + log(2)*(16*x + 8*x^2 + 8) - log(x)*(log(2)*(8*x + 8*x^2) + 4*x^2 + 4*x^3) + 8*x^2 + 4*x^ 3) + 6*x^2 + 6*x^3 + 2*x^4) - log(x/2 + log(2))*(exp(x)*(log(2)*(2*x + 6*x ^2 + 6*x^3 + 2*x^4) + x^2 + 3*x^3 + 3*x^4 + x^5) + log(x)^2*(16*x*log(2) + 8*x^2) - log(x)*(8*x + log(2)*(16*x + 16) + 8*x^2)) + log(x)*(4*x + 8*x^2 + 4*x^3))/(log(x/2 + log(2))*(log(2)*(2*x + 6*x^2 + 6*x^3 + 2*x^4) + x^2 + 3*x^3 + 3*x^4 + x^5)), x)
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.54 \[ \int \frac {\left (4 x+8 x^2+4 x^3\right ) \log (x)+\left (e^x \left (-x^2-3 x^3-3 x^4-x^5+\left (-2 x-6 x^2-6 x^3-2 x^4\right ) \log (2)\right )+\left (8 x+8 x^2+(16+16 x) \log (2)\right ) \log (x)+\left (-8 x^2-16 x \log (2)\right ) \log ^2(x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )+\left (2 x+6 x^2+6 x^3+2 x^4+\left (4 x+8 x^2+4 x^3+\left (8+16 x+8 x^2\right ) \log (2)+\left (-4 x^2-4 x^3+\left (-8 x-8 x^2\right ) \log (2)\right ) \log (x)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )\right ) \log \left (\log \left (\frac {1}{2} (x+2 \log (2))\right )\right )}{\left (x^2+3 x^3+3 x^4+x^5+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log (2)\right ) \log \left (\frac {1}{2} (x+2 \log (2))\right )} \, dx=\frac {-e^{x} x^{2}-2 e^{x} x -e^{x}+\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (2\right )+\frac {x}{2}\right )\right )^{2} x^{2}+2 \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (2\right )+\frac {x}{2}\right )\right )^{2} x +\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (2\right )+\frac {x}{2}\right )\right )^{2}+4 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (2\right )+\frac {x}{2}\right )\right ) \mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (2\right )+\frac {x}{2}\right )\right ) \mathrm {log}\left (x \right )+4 \mathrm {log}\left (x \right )^{2}}{x^{2}+2 x +1} \] Input:
int((((((-8*x^2-8*x)*log(2)-4*x^3-4*x^2)*log(x)+(8*x^2+16*x+8)*log(2)+4*x^ 3+8*x^2+4*x)*log(log(2)+1/2*x)+2*x^4+6*x^3+6*x^2+2*x)*log(log(log(2)+1/2*x ))+((-16*x*log(2)-8*x^2)*log(x)^2+((16*x+16)*log(2)+8*x^2+8*x)*log(x)+((-2 *x^4-6*x^3-6*x^2-2*x)*log(2)-x^5-3*x^4-3*x^3-x^2)*exp(x))*log(log(2)+1/2*x )+(4*x^3+8*x^2+4*x)*log(x))/((2*x^4+6*x^3+6*x^2+2*x)*log(2)+x^5+3*x^4+3*x^ 3+x^2)/log(log(2)+1/2*x),x)
Output:
( - e**x*x**2 - 2*e**x*x - e**x + log(log((2*log(2) + x)/2))**2*x**2 + 2*l og(log((2*log(2) + x)/2))**2*x + log(log((2*log(2) + x)/2))**2 + 4*log(log ((2*log(2) + x)/2))*log(x)*x + 4*log(log((2*log(2) + x)/2))*log(x) + 4*log (x)**2)/(x**2 + 2*x + 1)