Integrand size = 104, antiderivative size = 19 \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=1+\frac {4 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \] Output:
4*x^8/(ln(2*ln(2))-1+exp(x*ln(x^2)))^2+1
Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\frac {4 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \] Input:
Integrate[(-32*x^7 + (x^2)^x*(32*x^7 - 16*x^8 - 8*x^8*Log[x^2]) + 32*x^7*L og[Log[4]])/(-1 + (x^2)^(3*x) + 3*Log[Log[4]] - 3*Log[Log[4]]^2 + Log[Log[ 4]]^3 + (x^2)^(2*x)*(-3 + 3*Log[Log[4]]) + (x^2)^x*(3 - 6*Log[Log[4]] + 3* Log[Log[4]]^2)),x]
Output:
(4*x^8)/(-1 + (x^2)^x + Log[Log[4]])^2
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.
| \(\displaystyle \int \frac {-32 x^7+32 x^7 \log (\log (4))+\left (x^2\right )^x \left (-16 x^8+32 x^7-8 x^8 \log \left (x^2\right )\right )}{\left (x^2\right )^{3 x}+\left (x^2\right )^x \left (3+3 \log ^2(\log (4))-6 \log (\log (4))\right )+\left (x^2\right )^{2 x} (3 \log (\log (4))-3)-1+\log ^3(\log (4))-3 \log ^2(\log (4))+3 \log (\log (4))} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^7 (32 \log (\log (4))-32)+\left (x^2\right )^x \left (-16 x^8+32 x^7-8 x^8 \log \left (x^2\right )\right )}{\left (x^2\right )^{3 x}+\left (x^2\right )^x \left (3+3 \log ^2(\log (4))-6 \log (\log (4))\right )+\left (x^2\right )^{2 x} (3 \log (\log (4))-3)-1+\log ^3(\log (4))-3 \log ^2(\log (4))+3 \log (\log (4))}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 x^7 \left (2 x \left (x^2\right )^x+x \left (x^2\right )^x \log \left (x^2\right )-4 \left (\left (x^2\right )^x-1+\log (\log (4))\right )\right )}{\left (-\left (x^2\right )^x+1-\log (\log (4))\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \int \frac {x^7 \left (2 x \left (x^2\right )^x+x \log \left (x^2\right ) \left (x^2\right )^x+4 \left (-\left (x^2\right )^x-\log (\log (4))+1\right )\right )}{\left (-\left (x^2\right )^x-\log (\log (4))+1\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 8 \int \left (\frac {x^8 \left (\log \left (x^2\right )+2\right ) (-1+\log (\log (4)))}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^3}-\frac {x^7 \left (\log \left (x^2\right ) x+2 x-4\right )}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \left ((1-\log (\log (4))) \left (-\log \left (x^2\right )\right ) \int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^3}dx-2 (1-\log (\log (4))) \int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^3}dx-\log \left (x^2\right ) \int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^2}dx-2 \int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^2}dx+2 (1-\log (\log (4))) \int \frac {\int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^3}dx}{x}dx+2 \int \frac {\int \frac {x^8}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^2}dx}{x}dx+4 \int \frac {x^7}{\left (\left (x^2\right )^x+\log (\log (4))-1\right )^2}dx\right )\) |
Input:
Int[(-32*x^7 + (x^2)^x*(32*x^7 - 16*x^8 - 8*x^8*Log[x^2]) + 32*x^7*Log[Log [4]])/(-1 + (x^2)^(3*x) + 3*Log[Log[4]] - 3*Log[Log[4]]^2 + Log[Log[4]]^3 + (x^2)^(2*x)*(-3 + 3*Log[Log[4]]) + (x^2)^x*(3 - 6*Log[Log[4]] + 3*Log[Lo g[4]]^2)),x]
Output:
$Aborted
Time = 6.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {4 x^{8}}{{\left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+\left (x^{2}\right )^{x}-1\right )}^{2}}\) | \(20\) |
| parallelrisch | \(\frac {4 x^{8}}{\ln \left (2 \ln \left (2\right )\right )^{2}+2 \,{\mathrm e}^{x \ln \left (x^{2}\right )} \ln \left (2 \ln \left (2\right )\right )+{\mathrm e}^{2 x \ln \left (x^{2}\right )}-2 \ln \left (2 \ln \left (2\right )\right )-2 \,{\mathrm e}^{x \ln \left (x^{2}\right )}+1}\) | \(56\) |
Input:
int(((-8*x^8*ln(x^2)-16*x^8+32*x^7)*exp(x*ln(x^2))+32*x^7*ln(2*ln(2))-32*x ^7)/(exp(x*ln(x^2))^3+(3*ln(2*ln(2))-3)*exp(x*ln(x^2))^2+(3*ln(2*ln(2))^2- 6*ln(2*ln(2))+3)*exp(x*ln(x^2))+ln(2*ln(2))^3-3*ln(2*ln(2))^2+3*ln(2*ln(2) )-1),x,method=_RETURNVERBOSE)
Output:
4*x^8/(ln(2)+ln(ln(2))+(x^2)^x-1)^2
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.32 \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\frac {4 \, x^{8}}{2 \, {\left (x^{2}\right )}^{x} {\left (\log \left (2 \, \log \left (2\right )\right ) - 1\right )} + \log \left (2 \, \log \left (2\right )\right )^{2} + {\left (x^{2}\right )}^{2 \, x} - 2 \, \log \left (2 \, \log \left (2\right )\right ) + 1} \] Input:
integrate(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*lo g(2))-32*x^7)/(exp(x*log(x^2))^3+(3*log(2*log(2))-3)*exp(x*log(x^2))^2+(3* log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log(2 *log(2))^2+3*log(2*log(2))-1),x, algorithm="fricas")
Output:
4*x^8/(2*(x^2)^x*(log(2*log(2)) - 1) + log(2*log(2))^2 + (x^2)^(2*x) - 2*l og(2*log(2)) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\frac {4 x^{8}}{e^{2 x \log {\left (x^{2} \right )}} + \left (-2 + 2 \log {\left (\log {\left (2 \right )} \right )} + 2 \log {\left (2 \right )}\right ) e^{x \log {\left (x^{2} \right )}} - 2 \log {\left (2 \right )} + 2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \log {\left (\log {\left (2 \right )} \right )}^{2} + \log {\left (2 \right )}^{2} - 2 \log {\left (\log {\left (2 \right )} \right )} + 1} \] Input:
integrate(((-8*x**8*ln(x**2)-16*x**8+32*x**7)*exp(x*ln(x**2))+32*x**7*ln(2 *ln(2))-32*x**7)/(exp(x*ln(x**2))**3+(3*ln(2*ln(2))-3)*exp(x*ln(x**2))**2+ (3*ln(2*ln(2))**2-6*ln(2*ln(2))+3)*exp(x*ln(x**2))+ln(2*ln(2))**3-3*ln(2*l n(2))**2+3*ln(2*ln(2))-1),x)
Output:
4*x**8/(exp(2*x*log(x**2)) + (-2 + 2*log(log(2)) + 2*log(2))*exp(x*log(x** 2)) - 2*log(2) + 2*log(2)*log(log(2)) + log(log(2))**2 + log(2)**2 - 2*log (log(2)) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.68 \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\frac {4 \, x^{8}}{2 \, x^{2 \, x} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 1\right )} + 2 \, {\left (\log \left (\log \left (2\right )\right ) - 1\right )} \log \left (2\right ) + \log \left (2\right )^{2} + \log \left (\log \left (2\right )\right )^{2} + x^{4 \, x} - 2 \, \log \left (\log \left (2\right )\right ) + 1} \] Input:
integrate(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*lo g(2))-32*x^7)/(exp(x*log(x^2))^3+(3*log(2*log(2))-3)*exp(x*log(x^2))^2+(3* log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log(2 *log(2))^2+3*log(2*log(2))-1),x, algorithm="maxima")
Output:
4*x^8/(2*x^(2*x)*(log(2) + log(log(2)) - 1) + 2*(log(log(2)) - 1)*log(2) + log(2)^2 + log(log(2))^2 + x^(4*x) - 2*log(log(2)) + 1)
Timed out. \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\text {Timed out} \] Input:
integrate(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*lo g(2))-32*x^7)/(exp(x*log(x^2))^3+(3*log(2*log(2))-3)*exp(x*log(x^2))^2+(3* log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log(2 *log(2))^2+3*log(2*log(2))-1),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\text {Hanged} \] Input:
int(-(exp(x*log(x^2))*(8*x^8*log(x^2) - 32*x^7 + 16*x^8) - 32*x^7*log(2*lo g(2)) + 32*x^7)/(3*log(2*log(2)) + exp(3*x*log(x^2)) - 3*log(2*log(2))^2 + log(2*log(2))^3 + exp(x*log(x^2))*(3*log(2*log(2))^2 - 6*log(2*log(2)) + 3) + exp(2*x*log(x^2))*(3*log(2*log(2)) - 3) - 1),x)
Output:
\text{Hanged}
\[ \int \frac {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx=\int \frac {\left (-8 x^{8} \mathrm {log}\left (x^{2}\right )-16 x^{8}+32 x^{7}\right ) {\mathrm e}^{\mathrm {log}\left (x^{2}\right ) x}+32 x^{7} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )-32 x^{7}}{\left ({\mathrm e}^{\mathrm {log}\left (x^{2}\right ) x}\right )^{3}+\left (3 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )-3\right ) \left ({\mathrm e}^{\mathrm {log}\left (x^{2}\right ) x}\right )^{2}+\left (3 \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2}-6 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )+3\right ) {\mathrm e}^{\mathrm {log}\left (x^{2}\right ) x}+\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{3}-3 \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2}+3 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )-1}d x \] Input:
int(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*log(2))- 32*x^7)/(exp(x*log(x^2))^3+(3*log(2*log(2))-3)*exp(x*log(x^2))^2+(3*log(2* log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log(2*log(2 ))^2+3*log(2*log(2))-1),x)
Output:
int(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*log(2))- 32*x^7)/(exp(x*log(x^2))^3+(3*log(2*log(2))-3)*exp(x*log(x^2))^2+(3*log(2* log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log(2*log(2 ))^2+3*log(2*log(2))-1),x)