Integrand size = 89, antiderivative size = 24 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=-3+x \log (2) \left (x-\log ^2\left (e^{679}-\frac {\log (x)}{x}\right )\right ) \] Output:
x*ln(2)*(x-ln(exp(679)-ln(x)/x)^2)-3
\[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=\int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx \] Input:
Integrate[(-2*E^679*x^2*Log[2] + 2*x*Log[2]*Log[x] + (-2*Log[2] + 2*Log[2] *Log[x])*Log[(E^679*x - Log[x])/x] + (E^679*x*Log[2] - Log[2]*Log[x])*Log[ (E^679*x - Log[x])/x]^2)/(-(E^679*x) + Log[x]),x]
Output:
Integrate[(-2*E^679*x^2*Log[2] + 2*x*Log[2]*Log[x] + (-2*Log[2] + 2*Log[2] *Log[x])*Log[(E^679*x - Log[x])/x] + (E^679*x*Log[2] - Log[2]*Log[x])*Log[ (E^679*x - Log[x])/x]^2)/(-(E^679*x) + Log[x]), x]
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.
\(\displaystyle \int \frac {-2 e^{679} x^2 \log (2)+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )+2 x \log (2) \log (x)+(2 \log (2) \log (x)-2 \log (2)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )}{\log (x)-e^{679} x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\log (2) \log ^2\left (e^{679}-\frac {\log (x)}{x}\right )-\frac {2 \log (2) (\log (x)-1) \log \left (e^{679}-\frac {\log (x)}{x}\right )}{e^{679} x-\log (x)}+x \log (4)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log (2) \int \log ^2\left (e^{679}-\frac {\log (x)}{x}\right )dx+2 \log (2) \int \frac {\log \left (e^{679}-\frac {\log (x)}{x}\right )}{e^{679} x-\log (x)}dx-2 \log (2) \int \frac {\log (x) \log \left (e^{679}-\frac {\log (x)}{x}\right )}{e^{679} x-\log (x)}dx+\frac {1}{2} x^2 \log (4)\) |
Input:
Int[(-2*E^679*x^2*Log[2] + 2*x*Log[2]*Log[x] + (-2*Log[2] + 2*Log[2]*Log[x ])*Log[(E^679*x - Log[x])/x] + (E^679*x*Log[2] - Log[2]*Log[x])*Log[(E^679 *x - Log[x])/x]^2)/(-(E^679*x) + Log[x]),x]
Output:
$Aborted
Time = 1.64 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
method | result | size |
parallelrisch | \(-\ln \left (2\right ) x \ln \left (-\frac {\ln \left (x \right )-x \,{\mathrm e}^{679}}{x}\right )^{2}+x^{2} \ln \left (2\right )\) | \(29\) |
risch | \(\text {Expression too large to display}\) | \(676\) |
Input:
int(((-ln(2)*ln(x)+x*exp(679)*ln(2))*ln((-ln(x)+x*exp(679))/x)^2+(2*ln(2)* ln(x)-2*ln(2))*ln((-ln(x)+x*exp(679))/x)+2*x*ln(2)*ln(x)-2*x^2*exp(679)*ln (2))/(ln(x)-x*exp(679)),x,method=_RETURNVERBOSE)
Output:
-ln(2)*x*ln(-(ln(x)-x*exp(679))/x)^2+x^2*ln(2)
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=-x \log \left (2\right ) \log \left (\frac {x e^{679} - \log \left (x\right )}{x}\right )^{2} + x^{2} \log \left (2\right ) \] Input:
integrate(((-log(2)*log(x)+x*exp(679)*log(2))*log((-log(x)+x*exp(679))/x)^ 2+(2*log(2)*log(x)-2*log(2))*log((-log(x)+x*exp(679))/x)+2*x*log(2)*log(x) -2*x^2*exp(679)*log(2))/(log(x)-x*exp(679)),x, algorithm="fricas")
Output:
-x*log(2)*log((x*e^679 - log(x))/x)^2 + x^2*log(2)
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=x^{2} \log {\left (2 \right )} - x \log {\left (2 \right )} \log {\left (\frac {x e^{679} - \log {\left (x \right )}}{x} \right )}^{2} \] Input:
integrate(((-ln(2)*ln(x)+x*exp(679)*ln(2))*ln((-ln(x)+x*exp(679))/x)**2+(2 *ln(2)*ln(x)-2*ln(2))*ln((-ln(x)+x*exp(679))/x)+2*x*ln(2)*ln(x)-2*x**2*exp (679)*ln(2))/(ln(x)-x*exp(679)),x)
Output:
x**2*log(2) - x*log(2)*log((x*exp(679) - log(x))/x)**2
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=-x \log \left (2\right ) \log \left (x e^{679} - \log \left (x\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (x e^{679} - \log \left (x\right )\right ) \log \left (x\right ) - x \log \left (2\right ) \log \left (x\right )^{2} + x^{2} \log \left (2\right ) \] Input:
integrate(((-log(2)*log(x)+x*exp(679)*log(2))*log((-log(x)+x*exp(679))/x)^ 2+(2*log(2)*log(x)-2*log(2))*log((-log(x)+x*exp(679))/x)+2*x*log(2)*log(x) -2*x^2*exp(679)*log(2))/(log(x)-x*exp(679)),x, algorithm="maxima")
Output:
-x*log(2)*log(x*e^679 - log(x))^2 + 2*x*log(2)*log(x*e^679 - log(x))*log(x ) - x*log(2)*log(x)^2 + x^2*log(2)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=-x \log \left (2\right ) \log \left (x e^{679} - \log \left (x\right )\right )^{2} + 2 \, x \log \left (2\right ) \log \left (x e^{679} - \log \left (x\right )\right ) \log \left (x\right ) - x \log \left (2\right ) \log \left (x\right )^{2} + x^{2} \log \left (2\right ) \] Input:
integrate(((-log(2)*log(x)+x*exp(679)*log(2))*log((-log(x)+x*exp(679))/x)^ 2+(2*log(2)*log(x)-2*log(2))*log((-log(x)+x*exp(679))/x)+2*x*log(2)*log(x) -2*x^2*exp(679)*log(2))/(log(x)-x*exp(679)),x, algorithm="giac")
Output:
-x*log(2)*log(x*e^679 - log(x))^2 + 2*x*log(2)*log(x*e^679 - log(x))*log(x ) - x*log(2)*log(x)^2 + x^2*log(2)
Time = 4.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=x\,\ln \left (2\right )\,\left (x-{\ln \left (-\frac {\ln \left (x\right )-x\,{\mathrm {e}}^{679}}{x}\right )}^2\right ) \] Input:
int(-(log(-(log(x) - x*exp(679))/x)*(2*log(2) - 2*log(2)*log(x)) + log(-(l og(x) - x*exp(679))/x)^2*(log(2)*log(x) - x*exp(679)*log(2)) + 2*x^2*exp(6 79)*log(2) - 2*x*log(2)*log(x))/(log(x) - x*exp(679)),x)
Output:
x*log(2)*(x - log(-(log(x) - x*exp(679))/x)^2)
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e^{679} x^2 \log (2)+2 x \log (2) \log (x)+(-2 \log (2)+2 \log (2) \log (x)) \log \left (\frac {e^{679} x-\log (x)}{x}\right )+\left (e^{679} x \log (2)-\log (2) \log (x)\right ) \log ^2\left (\frac {e^{679} x-\log (x)}{x}\right )}{-e^{679} x+\log (x)} \, dx=\mathrm {log}\left (2\right ) x \left (-\mathrm {log}\left (\frac {-\mathrm {log}\left (x \right )+e^{679} x}{x}\right )^{2}+x \right ) \] Input:
int(((-log(2)*log(x)+x*exp(679)*log(2))*log((-log(x)+x*exp(679))/x)^2+(2*l og(2)*log(x)-2*log(2))*log((-log(x)+x*exp(679))/x)+2*x*log(2)*log(x)-2*x^2 *exp(679)*log(2))/(log(x)-x*exp(679)),x)
Output:
log(2)*x*( - log(( - log(x) + e**679*x)/x)**2 + x)