Integrand size = 102, antiderivative size = 22 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=x \left (4 x-x \log \left (-x+\log \left (x-x^4\right )\right )\right ) \] Output:
(-ln(ln(-x^4+x)-x)*x+4*x)*x
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=4 x^2-x^2 \log \left (-x+\log \left (x-x^4\right )\right ) \] Input:
Integrate[(x + 7*x^2 - 4*x^4 - 7*x^5 + (-8*x + 8*x^4)*Log[x - x^4] + (-2*x ^2 + 2*x^5 + (2*x - 2*x^4)*Log[x - x^4])*Log[-x + Log[x - x^4]])/(x - x^4 + (-1 + x^3)*Log[x - x^4]),x]
Output:
4*x^2 - x^2*Log[-x + Log[x - x^4]]
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.
\(\displaystyle \int \frac {-7 x^5-4 x^4+\left (8 x^4-8 x\right ) \log \left (x-x^4\right )+7 x^2+\left (2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )-2 x^2\right ) \log \left (\log \left (x-x^4\right )-x\right )+x}{-x^4+\left (x^3-1\right ) \log \left (x-x^4\right )+x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-7 x^5-4 x^4+\left (8 x^4-8 x\right ) \log \left (x-x^4\right )+7 x^2+\left (2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )-2 x^2\right ) \log \left (\log \left (x-x^4\right )-x\right )+x}{\left (1-x^3\right ) \left (x-\log \left (x-x^4\right )\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-2 x \log \left (\log \left (x-x^4\right )-x\right )+\frac {8 x \log \left (x \left (1-x^3\right )\right )}{\log \left (x-x^4\right )-x}+\frac {4 x^4}{(x-1) \left (x^2+x+1\right ) \left (x-\log \left (x-x^4\right )\right )}-\frac {7 x^2}{(x-1) \left (x^2+x+1\right ) \left (x-\log \left (x-x^4\right )\right )}-\frac {x}{(x-1) \left (x^2+x+1\right ) \left (x-\log \left (x-x^4\right )\right )}+\frac {7 x^5}{(x-1) \left (x^2+x+1\right ) \left (x-\log \left (x-x^4\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 i \int \frac {1}{\left (-2 x+i \sqrt {3}-1\right ) \left (x-\log \left (x-x^4\right )\right )}dx}{\sqrt {3}}+\int \frac {1}{(x-1) \left (x-\log \left (x-x^4\right )\right )}dx+4 \int \frac {x}{x-\log \left (x-x^4\right )}dx-\frac {1}{3} \left (3+i \sqrt {3}\right ) \int \frac {1}{\left (2 x-i \sqrt {3}+1\right ) \left (x-\log \left (x-x^4\right )\right )}dx-\frac {1}{3} \left (3-i \sqrt {3}\right ) \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \left (x-\log \left (x-x^4\right )\right )}dx+\frac {2 i \int \frac {1}{\left (2 x+i \sqrt {3}+1\right ) \left (x-\log \left (x-x^4\right )\right )}dx}{\sqrt {3}}-2 \int x \log \left (\log \left (x-x^4\right )-x\right )dx+8 \int \frac {x \log \left (x \left (1-x^3\right )\right )}{\log \left (x-x^4\right )-x}dx+7 \int \frac {x^2}{x-\log \left (x-x^4\right )}dx\) |
Input:
Int[(x + 7*x^2 - 4*x^4 - 7*x^5 + (-8*x + 8*x^4)*Log[x - x^4] + (-2*x^2 + 2 *x^5 + (2*x - 2*x^4)*Log[x - x^4])*Log[-x + Log[x - x^4]])/(x - x^4 + (-1 + x^3)*Log[x - x^4]),x]
Output:
$Aborted
Time = 4.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(-\ln \left (\ln \left (-x^{4}+x \right )-x \right ) x^{2}+4 x^{2}\) | \(25\) |
Input:
int((((-2*x^4+2*x)*ln(-x^4+x)+2*x^5-2*x^2)*ln(ln(-x^4+x)-x)+(8*x^4-8*x)*ln (-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-1)*ln(-x^4+x)-x^4+x),x,method=_RETURNV ERBOSE)
Output:
-ln(ln(-x^4+x)-x)*x^2+4*x^2
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=-x^{2} \log \left (-x + \log \left (-x^{4} + x\right )\right ) + 4 \, x^{2} \] Input:
integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^ 4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, alg orithm="fricas")
Output:
-x^2*log(-x + log(-x^4 + x)) + 4*x^2
Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=- x^{2} \log {\left (- x + \log {\left (- x^{4} + x \right )} \right )} + 4 x^{2} \] Input:
integrate((((-2*x**4+2*x)*ln(-x**4+x)+2*x**5-2*x**2)*ln(ln(-x**4+x)-x)+(8* x**4-8*x)*ln(-x**4+x)-7*x**5-4*x**4+7*x**2+x)/((x**3-1)*ln(-x**4+x)-x**4+x ),x)
Output:
-x**2*log(-x + log(-x**4 + x)) + 4*x**2
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=-x^{2} \log \left (-x + \log \left (x^{2} + x + 1\right ) + \log \left (x\right ) + \log \left (-x + 1\right )\right ) + 4 \, x^{2} \] Input:
integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^ 4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, alg orithm="maxima")
Output:
-x^2*log(-x + log(x^2 + x + 1) + log(x) + log(-x + 1)) + 4*x^2
Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=-x^{2} \log \left (-x + \log \left (-x^{4} + x\right )\right ) + 4 \, x^{2} \] Input:
integrate((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^ 4-8*x)*log(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-1)*log(-x^4+x)-x^4+x),x, alg orithm="giac")
Output:
-x^2*log(-x + log(-x^4 + x)) + 4*x^2
Time = 1.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=-x^2\,\left (\ln \left (\ln \left (x-x^4\right )-x\right )-4\right ) \] Input:
int((x - log(x - x^4)*(8*x - 8*x^4) + log(log(x - x^4) - x)*(log(x - x^4)* (2*x - 2*x^4) - 2*x^2 + 2*x^5) + 7*x^2 - 4*x^4 - 7*x^5)/(x + log(x - x^4)* (x^3 - 1) - x^4),x)
Output:
-x^2*(log(log(x - x^4) - x) - 4)
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {x+7 x^2-4 x^4-7 x^5+\left (-8 x+8 x^4\right ) \log \left (x-x^4\right )+\left (-2 x^2+2 x^5+\left (2 x-2 x^4\right ) \log \left (x-x^4\right )\right ) \log \left (-x+\log \left (x-x^4\right )\right )}{x-x^4+\left (-1+x^3\right ) \log \left (x-x^4\right )} \, dx=x^{2} \left (-\mathrm {log}\left (\mathrm {log}\left (-x^{4}+x \right )-x \right )+4\right ) \] Input:
int((((-2*x^4+2*x)*log(-x^4+x)+2*x^5-2*x^2)*log(log(-x^4+x)-x)+(8*x^4-8*x) *log(-x^4+x)-7*x^5-4*x^4+7*x^2+x)/((x^3-1)*log(-x^4+x)-x^4+x),x)
Output:
x**2*( - log(log( - x**4 + x) - x) + 4)