Integrand size = 97, antiderivative size = 25 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=x \log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \] Output:
x*ln(x/(2*x*ln(ln(x^4))+2*exp(x)-2*x-8))
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=x \log \left (\frac {x}{2 \left (-4+e^x-x+x \log \left (\log \left (x^4\right )\right )\right )}\right ) \] Input:
Integrate[(-4*x + (-4 + E^x*(1 - x))*Log[x^4] + ((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]])*Log[x/(-8 + 2*E^x - 2*x + 2*x*Log[Log[x^4]])])/ ((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]]),x]
Output:
x*Log[x/(2*(-4 + E^x - x + x*Log[Log[x^4]]))]
Time = 2.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7292, 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.
\(\displaystyle \int \frac {\left (e^x (1-x)-4\right ) \log \left (x^4\right )+\left (\left (-x+e^x-4\right ) \log \left (x^4\right )+x \log \left (\log \left (x^4\right )\right ) \log \left (x^4\right )\right ) \log \left (\frac {x}{2 x \log \left (\log \left (x^4\right )\right )-2 x+2 e^x-8}\right )-4 x}{\left (-x+e^x-4\right ) \log \left (x^4\right )+x \log \left (\log \left (x^4\right )\right ) \log \left (x^4\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (e^x (1-x)-4\right ) \log \left (x^4\right )-\left (\left (-x+e^x-4\right ) \log \left (x^4\right )+x \log \left (\log \left (x^4\right )\right ) \log \left (x^4\right )\right ) \log \left (\frac {x}{2 x \log \left (\log \left (x^4\right )\right )-2 x+2 e^x-8}\right )+4 x}{\log \left (x^4\right ) \left (x \left (-\log \left (\log \left (x^4\right )\right )\right )+x-e^x+4\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x \left (-x \log \left (x^4\right )+x \log \left (\log \left (x^4\right )\right ) \log \left (x^4\right )-\log \left (\log \left (x^4\right )\right ) \log \left (x^4\right )-3 \log \left (x^4\right )-4\right )}{\log \left (x^4\right ) \left (x \log \left (\log \left (x^4\right )\right )-x+e^x-4\right )}+\log \left (\frac {x}{2 \left (x \log \left (\log \left (x^4\right )\right )-x+e^x-4\right )}\right )-x+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \log \left (-\frac {x}{2 \left (x \left (-\log \left (\log \left (x^4\right )\right )\right )+x-e^x+4\right )}\right )\) |
Input:
Int[(-4*x + (-4 + E^x*(1 - x))*Log[x^4] + ((-4 + E^x - x)*Log[x^4] + x*Log [x^4]*Log[Log[x^4]])*Log[x/(-8 + 2*E^x - 2*x + 2*x*Log[Log[x^4]])])/((-4 + E^x - x)*Log[x^4] + x*Log[x^4]*Log[Log[x^4]]),x]
Output:
x*Log[-1/2*x/(4 - E^x + x - x*Log[Log[x^4]])]
Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(24)=48\).
Time = 25.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68
method | result | size |
parallelrisch | \(x \ln \left (\frac {x}{2 x \ln \left (\ln \left (x^{4}\right )\right )+2 \,{\mathrm e}^{x}-2 x -8}\right )+4 \ln \left (x \right )-4 \ln \left (x \ln \left (\ln \left (x^{4}\right )\right )+{\mathrm e}^{x}-x -4\right )-4 \ln \left (\frac {x}{2 x \ln \left (\ln \left (x^{4}\right )\right )+2 \,{\mathrm e}^{x}-2 x -8}\right )\) | \(67\) |
risch | \(\text {Expression too large to display}\) | \(994\) |
Input:
int(((x*ln(x^4)*ln(ln(x^4))+(exp(x)-4-x)*ln(x^4))*ln(x/(2*x*ln(ln(x^4))+2* exp(x)-2*x-8))+((1-x)*exp(x)-4)*ln(x^4)-4*x)/(x*ln(x^4)*ln(ln(x^4))+(exp(x )-4-x)*ln(x^4)),x,method=_RETURNVERBOSE)
Output:
ln(1/2*x/(x*ln(ln(x^4))+exp(x)-x-4))*x+4*ln(x)-4*ln(x*ln(ln(x^4))+exp(x)-x -4)-4*ln(1/2*x/(x*ln(ln(x^4))+exp(x)-x-4))
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=x \log \left (\frac {x}{2 \, {\left (x \log \left (\log \left (x^{4}\right )\right ) - x + e^{x} - 4\right )}}\right ) \] Input:
integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log (log(x^4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log (log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="fricas")
Output:
x*log(1/2*x/(x*log(log(x^4)) - x + e^x - 4))
Time = 7.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=x \log {\left (\frac {x}{2 x \log {\left (\log {\left (x^{4} \right )} \right )} - 2 x + 2 e^{x} - 8} \right )} \] Input:
integrate(((x*ln(x**4)*ln(ln(x**4))+(exp(x)-4-x)*ln(x**4))*ln(x/(2*x*ln(ln (x**4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*ln(x**4)-4*x)/(x*ln(x**4)*ln(ln( x**4))+(exp(x)-4-x)*ln(x**4)),x)
Output:
x*log(x/(2*x*log(log(x**4)) - 2*x + 2*exp(x) - 8))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=-x \log \left (2\right ) - x \log \left (x {\left (2 \, \log \left (2\right ) - 1\right )} + x \log \left (\log \left (x\right )\right ) + e^{x} - 4\right ) + x \log \left (x\right ) \] Input:
integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log (log(x^4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log (log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="maxima")
Output:
-x*log(2) - x*log(x*(2*log(2) - 1) + x*log(log(x)) + e^x - 4) + x*log(x)
\[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=\int { -\frac {{\left ({\left (x - 1\right )} e^{x} + 4\right )} \log \left (x^{4}\right ) - {\left (x \log \left (x^{4}\right ) \log \left (\log \left (x^{4}\right )\right ) - {\left (x - e^{x} + 4\right )} \log \left (x^{4}\right )\right )} \log \left (\frac {x}{2 \, {\left (x \log \left (\log \left (x^{4}\right )\right ) - x + e^{x} - 4\right )}}\right ) + 4 \, x}{x \log \left (x^{4}\right ) \log \left (\log \left (x^{4}\right )\right ) - {\left (x - e^{x} + 4\right )} \log \left (x^{4}\right )} \,d x } \] Input:
integrate(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log (log(x^4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log (log(x^4))+(exp(x)-4-x)*log(x^4)),x, algorithm="giac")
Output:
integrate(-(((x - 1)*e^x + 4)*log(x^4) - (x*log(x^4)*log(log(x^4)) - (x - e^x + 4)*log(x^4))*log(1/2*x/(x*log(log(x^4)) - x + e^x - 4)) + 4*x)/(x*lo g(x^4)*log(log(x^4)) - (x - e^x + 4)*log(x^4)), x)
Time = 2.88 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=x\,\ln \left (-\frac {x}{2\,x-2\,{\mathrm {e}}^x-2\,x\,\ln \left (\ln \left (x^4\right )\right )+8}\right ) \] Input:
int((4*x + log(x^4)*(exp(x)*(x - 1) + 4) + log(-x/(2*x - 2*exp(x) - 2*x*lo g(log(x^4)) + 8))*(log(x^4)*(x - exp(x) + 4) - x*log(x^4)*log(log(x^4))))/ (log(x^4)*(x - exp(x) + 4) - x*log(x^4)*log(log(x^4))),x)
Output:
x*log(-x/(2*x - 2*exp(x) - 2*x*log(log(x^4)) + 8))
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x+\left (-4+e^x (1-x)\right ) \log \left (x^4\right )+\left (\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )\right ) \log \left (\frac {x}{-8+2 e^x-2 x+2 x \log \left (\log \left (x^4\right )\right )}\right )}{\left (-4+e^x-x\right ) \log \left (x^4\right )+x \log \left (x^4\right ) \log \left (\log \left (x^4\right )\right )} \, dx=\mathrm {log}\left (\frac {x}{2 e^{x}+2 \,\mathrm {log}\left (\mathrm {log}\left (x^{4}\right )\right ) x -2 x -8}\right ) x \] Input:
int(((x*log(x^4)*log(log(x^4))+(exp(x)-4-x)*log(x^4))*log(x/(2*x*log(log(x ^4))+2*exp(x)-2*x-8))+((1-x)*exp(x)-4)*log(x^4)-4*x)/(x*log(x^4)*log(log(x ^4))+(exp(x)-4-x)*log(x^4)),x)
Output:
log(x/(2*e**x + 2*log(log(x**4))*x - 2*x - 8))*x