Integrand size = 86, antiderivative size = 28 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\frac {(4-x) x^2 \log \left (\log ^4\left ((1-x)^2\right )\right )}{5 (1+x)} \] Output:
1/5*x^2/(1+x)*(4-x)*ln(ln((1-x)^2)^4)
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\frac {1}{5} \left (-20 \log \left (\log \left ((-1+x)^2\right )\right )-\frac {\left (-5-5 x-4 x^2+x^3\right ) \log \left (\log ^4\left ((-1+x)^2\right )\right )}{1+x}\right ) \] Input:
Integrate[(32*x^2 + 24*x^3 - 8*x^4 + (-8*x + 7*x^2 + 3*x^3 - 2*x^4)*Log[1 - 2*x + x^2]*Log[Log[1 - 2*x + x^2]^4])/((-5 - 5*x + 5*x^2 + 5*x^3)*Log[1 - 2*x + x^2]),x]
Output:
(-20*Log[Log[(-1 + x)^2]] - ((-5 - 5*x - 4*x^2 + x^3)*Log[Log[(-1 + x)^2]^ 4])/(1 + x))/5
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.
\(\displaystyle \int \frac {-8 x^4+24 x^3+32 x^2+\left (-2 x^4+3 x^3+7 x^2-8 x\right ) \log \left (x^2-2 x+1\right ) \log \left (\log ^4\left (x^2-2 x+1\right )\right )}{\left (5 x^3+5 x^2-5 x-5\right ) \log \left (x^2-2 x+1\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-8 x^4+24 x^3+32 x^2+\left (-2 x^4+3 x^3+7 x^2-8 x\right ) \log \left (x^2-2 x+1\right ) \log \left (\log ^4\left (x^2-2 x+1\right )\right )}{10 \left (x^2-1\right ) \log \left (x^2-2 x+1\right )}-\frac {-8 x^4+24 x^3+32 x^2+\left (-2 x^4+3 x^3+7 x^2-8 x\right ) \log \left (x^2-2 x+1\right ) \log \left (\log ^4\left (x^2-2 x+1\right )\right )}{10 (x+1)^2 \log \left (x^2-2 x+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{5} \int x \log \left (\log ^4\left ((x-1)^2\right )\right )dx-\int \frac {\log \left (\log ^4\left ((x-1)^2\right )\right )}{(x+1)^2}dx-4 \int \frac {1}{(x+1) \log \left ((x-1)^2\right )}dx+\frac {8 (1-x) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left ((x-1)^2\right )\right )}{5 \sqrt {(x-1)^2}}-\frac {4}{5} \operatorname {LogIntegral}\left ((x-1)^2\right )-(1-x) \log \left (\log ^4\left ((x-1)^2\right )\right )+\frac {6}{5} \log \left (\log \left ((x-1)^2\right )\right )\) |
Input:
Int[(32*x^2 + 24*x^3 - 8*x^4 + (-8*x + 7*x^2 + 3*x^3 - 2*x^4)*Log[1 - 2*x + x^2]*Log[Log[1 - 2*x + x^2]^4])/((-5 - 5*x + 5*x^2 + 5*x^3)*Log[1 - 2*x + x^2]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(26)=52\).
Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46
method | result | size |
parallelrisch | \(\frac {-x^{3} \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )+4 x^{2} \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )-16 \ln \left (\ln \left (x^{2}-2 x +1\right )\right ) x +4 x \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )-16 \ln \left (\ln \left (x^{2}-2 x +1\right )\right )+4 \ln \left (\ln \left (x^{2}-2 x +1\right )^{4}\right )}{5 x +5}\) | \(97\) |
Input:
int(((-2*x^4+3*x^3+7*x^2-8*x)*ln(x^2-2*x+1)*ln(ln(x^2-2*x+1)^4)-8*x^4+24*x ^3+32*x^2)/(5*x^3+5*x^2-5*x-5)/ln(x^2-2*x+1),x,method=_RETURNVERBOSE)
Output:
1/5*(-x^3*ln(ln(x^2-2*x+1)^4)+4*x^2*ln(ln(x^2-2*x+1)^4)-16*ln(ln(x^2-2*x+1 ))*x+4*x*ln(ln(x^2-2*x+1)^4)-16*ln(ln(x^2-2*x+1))+4*ln(ln(x^2-2*x+1)^4))/( 1+x)
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {{\left (x^{3} - 4 \, x^{2}\right )} \log \left (\log \left (x^{2} - 2 \, x + 1\right )^{4}\right )}{5 \, {\left (x + 1\right )}} \] Input:
integrate(((-2*x^4+3*x^3+7*x^2-8*x)*log(x^2-2*x+1)*log(log(x^2-2*x+1)^4)-8 *x^4+24*x^3+32*x^2)/(5*x^3+5*x^2-5*x-5)/log(x^2-2*x+1),x, algorithm="frica s")
Output:
-1/5*(x^3 - 4*x^2)*log(log(x^2 - 2*x + 1)^4)/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=- 4 \log {\left (\log {\left (x^{2} - 2 x + 1 \right )} \right )} + \frac {\left (- x^{3} + 4 x^{2} + 5 x + 5\right ) \log {\left (\log {\left (x^{2} - 2 x + 1 \right )}^{4} \right )}}{5 x + 5} \] Input:
integrate(((-2*x**4+3*x**3+7*x**2-8*x)*ln(x**2-2*x+1)*ln(ln(x**2-2*x+1)**4 )-8*x**4+24*x**3+32*x**2)/(5*x**3+5*x**2-5*x-5)/ln(x**2-2*x+1),x)
Output:
-4*log(log(x**2 - 2*x + 1)) + (-x**3 + 4*x**2 + 5*x + 5)*log(log(x**2 - 2* x + 1)**4)/(5*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {4 \, {\left (x^{3} \log \left (2\right ) - 4 \, x^{2} \log \left (2\right ) - 5 \, x \log \left (2\right ) + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (\log \left (x - 1\right )\right ) - 5 \, \log \left (2\right )\right )}}{5 \, {\left (x + 1\right )}} \] Input:
integrate(((-2*x^4+3*x^3+7*x^2-8*x)*log(x^2-2*x+1)*log(log(x^2-2*x+1)^4)-8 *x^4+24*x^3+32*x^2)/(5*x^3+5*x^2-5*x-5)/log(x^2-2*x+1),x, algorithm="maxim a")
Output:
-4/5*(x^3*log(2) - 4*x^2*log(2) - 5*x*log(2) + (x^3 - 4*x^2)*log(log(x - 1 )) - 5*log(2))/(x + 1)
Time = 0.46 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {1}{5} \, {\left (x^{2} - 5 \, x - \frac {5}{x + 1}\right )} \log \left (\log \left (x^{2} - 2 \, x + 1\right )^{4}\right ) - 4 \, \log \left (\log \left (x^{2} - 2 \, x + 1\right )\right ) \] Input:
integrate(((-2*x^4+3*x^3+7*x^2-8*x)*log(x^2-2*x+1)*log(log(x^2-2*x+1)^4)-8 *x^4+24*x^3+32*x^2)/(5*x^3+5*x^2-5*x-5)/log(x^2-2*x+1),x, algorithm="giac" )
Output:
-1/5*(x^2 - 5*x - 5/(x + 1))*log(log(x^2 - 2*x + 1)^4) - 4*log(log(x^2 - 2 *x + 1))
Time = 4.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=-\frac {x^2\,\ln \left ({\ln \left (x^2-2\,x+1\right )}^4\right )\,\left (x-4\right )}{5\,\left (x+1\right )} \] Input:
int(-(32*x^2 + 24*x^3 - 8*x^4 - log(log(x^2 - 2*x + 1)^4)*log(x^2 - 2*x + 1)*(8*x - 7*x^2 - 3*x^3 + 2*x^4))/(log(x^2 - 2*x + 1)*(5*x - 5*x^2 - 5*x^3 + 5)),x)
Output:
-(x^2*log(log(x^2 - 2*x + 1)^4)*(x - 4))/(5*(x + 1))
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {32 x^2+24 x^3-8 x^4+\left (-8 x+7 x^2+3 x^3-2 x^4\right ) \log \left (1-2 x+x^2\right ) \log \left (\log ^4\left (1-2 x+x^2\right )\right )}{\left (-5-5 x+5 x^2+5 x^3\right ) \log \left (1-2 x+x^2\right )} \, dx=\frac {-\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )^{4}\right ) x^{3}+4 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )^{4}\right ) x^{2}-4 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )^{4}\right ) x -4 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )^{4}\right )+16 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )\right ) x +16 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-2 x +1\right )\right )}{5 x +5} \] Input:
int(((-2*x^4+3*x^3+7*x^2-8*x)*log(x^2-2*x+1)*log(log(x^2-2*x+1)^4)-8*x^4+2 4*x^3+32*x^2)/(5*x^3+5*x^2-5*x-5)/log(x^2-2*x+1),x)
Output:
( - log(log(x**2 - 2*x + 1)**4)*x**3 + 4*log(log(x**2 - 2*x + 1)**4)*x**2 - 4*log(log(x**2 - 2*x + 1)**4)*x - 4*log(log(x**2 - 2*x + 1)**4) + 16*log (log(x**2 - 2*x + 1))*x + 16*log(log(x**2 - 2*x + 1)))/(5*(x + 1))