Integrand size = 65, antiderivative size = 26 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=3-3 x-\frac {1}{4} (10-x) x \log \left (-1+x+\frac {x^4}{4}\right ) \] Output:
3-3*x-1/4*x*ln(1/4*x^4+x-1)*(10-x)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=\frac {1}{2} \left (-6 x+\frac {1}{2} (-10+x) x \log \left (-1+x+\frac {x^4}{4}\right )\right ) \] Input:
Integrate[(24 - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + (20 - 24*x + 4*x^2 - 5*x^4 + x^5)*Log[(-4 + 4*x + x^4)/4])/(-8 + 8*x + 2*x^4),x]
Output:
(-6*x + ((-10 + x)*x*Log[-1 + x + x^4/4])/2)/2
Time = 0.66 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {2 x^5-26 x^4+2 x^2+\left (x^5-5 x^4+4 x^2-24 x+20\right ) \log \left (\frac {1}{4} \left (x^4+4 x-4\right )\right )-44 x+24}{2 x^4+8 x-8} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{2} (x-5) \log \left (\frac {x^4}{4}+x-1\right )+\frac {x^5-13 x^4+x^2-22 x+12}{x^4+4 x-4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} (5-x)^2 \log \left (\frac {x^4}{4}+x-1\right )-\frac {25}{4} \log \left (-x^4-4 x+4\right )-3 x\) |
Input:
Int[(24 - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + (20 - 24*x + 4*x^2 - 5*x^4 + x^5 )*Log[(-4 + 4*x + x^4)/4])/(-8 + 8*x + 2*x^4),x]
Output:
-3*x - (25*Log[4 - 4*x - x^4])/4 + ((5 - x)^2*Log[-1 + x + x^4/4])/4
Time = 0.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\left (-\frac {5}{2} x +\frac {1}{4} x^{2}\right ) \ln \left (\frac {1}{4} x^{4}+x -1\right )-3 x\) | \(24\) |
norman | \(-3 x -\frac {5 \ln \left (\frac {1}{4} x^{4}+x -1\right ) x}{2}+\frac {\ln \left (\frac {1}{4} x^{4}+x -1\right ) x^{2}}{4}\) | \(31\) |
parallelrisch | \(-3 x -\frac {5 \ln \left (\frac {1}{4} x^{4}+x -1\right ) x}{2}+\frac {\ln \left (\frac {1}{4} x^{4}+x -1\right ) x^{2}}{4}\) | \(31\) |
default | \(\frac {x^{2} \ln \left (x^{4}+4 x -4\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (3 \textit {\_R}^{2}-4 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}-\frac {5 x \ln \left (x^{4}+4 x -4\right )}{2}-3 x +\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{2}+\ln \left (2\right ) \left (5 x -\frac {1}{2} x^{2}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+34 \textit {\_R} -40\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}\) | \(151\) |
parts | \(\frac {x^{2} \ln \left (x^{4}+4 x -4\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (3 \textit {\_R}^{2}-4 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}-\frac {x^{2} \ln \left (2\right )}{2}+5 x \ln \left (2\right )-\frac {5 x \ln \left (x^{4}+4 x -4\right )}{2}-3 x +\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R} +4\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z} -4\right )}{\sum }\frac {\left (-3 \textit {\_R}^{2}+34 \textit {\_R} -40\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+1}\right )}{4}\) | \(151\) |
orering | \(\frac {\left (3 x^{10}-34 x^{9}+24 x^{7}-360 x^{6}+1032 x^{5}-1188 x^{4}-80 x^{3}-1152 x^{2}+2496 x -1600\right ) \left (\left (x^{5}-5 x^{4}+4 x^{2}-24 x +20\right ) \ln \left (\frac {1}{4} x^{4}+x -1\right )+2 x^{5}-26 x^{4}+2 x^{2}-44 x +24\right )}{2 \left (2 x^{9}-12 x^{8}+50 x^{7}+19 x^{6}-210 x^{5}+976 x^{4}-992 x^{3}-24 x^{2}+224 x -352\right ) \left (2 x^{4}+8 x -8\right )}-\frac {\left (x^{7}-17 x^{6}+501 x^{4}-8 x^{3}-312 x^{2}+2400 x -2000\right ) \left (x^{4}+4 x -4\right ) \left (\frac {\left (5 x^{4}-20 x^{3}+8 x -24\right ) \ln \left (\frac {1}{4} x^{4}+x -1\right )+\frac {\left (x^{5}-5 x^{4}+4 x^{2}-24 x +20\right ) \left (x^{3}+1\right )}{\frac {1}{4} x^{4}+x -1}+10 x^{4}-104 x^{3}+4 x -44}{2 x^{4}+8 x -8}-\frac {\left (\left (x^{5}-5 x^{4}+4 x^{2}-24 x +20\right ) \ln \left (\frac {1}{4} x^{4}+x -1\right )+2 x^{5}-26 x^{4}+2 x^{2}-44 x +24\right ) \left (8 x^{3}+8\right )}{\left (2 x^{4}+8 x -8\right )^{2}}\right )}{2 \left (2 x^{9}-12 x^{8}+50 x^{7}+19 x^{6}-210 x^{5}+976 x^{4}-992 x^{3}-24 x^{2}+224 x -352\right )}\) | \(398\) |
Input:
int(((x^5-5*x^4+4*x^2-24*x+20)*ln(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24) /(2*x^4+8*x-8),x,method=_RETURNVERBOSE)
Output:
(-5/2*x+1/4*x^2)*ln(1/4*x^4+x-1)-3*x
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=\frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (\frac {1}{4} \, x^{4} + x - 1\right ) - 3 \, x \] Input:
integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-4 4*x+24)/(2*x^4+8*x-8),x, algorithm="fricas")
Output:
1/4*(x^2 - 10*x)*log(1/4*x^4 + x - 1) - 3*x
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=- 3 x + \left (\frac {x^{2}}{4} - \frac {5 x}{2}\right ) \log {\left (\frac {x^{4}}{4} + x - 1 \right )} \] Input:
integrate(((x**5-5*x**4+4*x**2-24*x+20)*ln(1/4*x**4+x-1)+2*x**5-26*x**4+2* x**2-44*x+24)/(2*x**4+8*x-8),x)
Output:
-3*x + (x**2/4 - 5*x/2)*log(x**4/4 + x - 1)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=-\frac {1}{2} \, x^{2} \log \left (2\right ) + x {\left (5 \, \log \left (2\right ) - 3\right )} + \frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (x^{4} + 4 \, x - 4\right ) \] Input:
integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-4 4*x+24)/(2*x^4+8*x-8),x, algorithm="maxima")
Output:
-1/2*x^2*log(2) + x*(5*log(2) - 3) + 1/4*(x^2 - 10*x)*log(x^4 + 4*x - 4)
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=\frac {1}{4} \, {\left (x^{2} - 10 \, x\right )} \log \left (\frac {1}{4} \, x^{4} + x - 1\right ) - 3 \, x \] Input:
integrate(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-4 4*x+24)/(2*x^4+8*x-8),x, algorithm="giac")
Output:
1/4*(x^2 - 10*x)*log(1/4*x^4 + x - 1) - 3*x
Time = 3.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=-3\,x-\ln \left (\frac {x^4}{4}+x-1\right )\,\left (\frac {5\,x}{2}-\frac {x^2}{4}\right ) \] Input:
int((log(x + x^4/4 - 1)*(4*x^2 - 24*x - 5*x^4 + x^5 + 20) - 44*x + 2*x^2 - 26*x^4 + 2*x^5 + 24)/(8*x + 2*x^4 - 8),x)
Output:
- 3*x - log(x + x^4/4 - 1)*((5*x)/2 - x^2/4)
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {24-44 x+2 x^2-26 x^4+2 x^5+\left (20-24 x+4 x^2-5 x^4+x^5\right ) \log \left (\frac {1}{4} \left (-4+4 x+x^4\right )\right )}{-8+8 x+2 x^4} \, dx=\frac {x \left (\mathrm {log}\left (\frac {1}{4} x^{4}+x -1\right ) x -10 \,\mathrm {log}\left (\frac {1}{4} x^{4}+x -1\right )-12\right )}{4} \] Input:
int(((x^5-5*x^4+4*x^2-24*x+20)*log(1/4*x^4+x-1)+2*x^5-26*x^4+2*x^2-44*x+24 )/(2*x^4+8*x-8),x)
Output:
(x*(log((x**4 + 4*x - 4)/4)*x - 10*log((x**4 + 4*x - 4)/4) - 12))/4