Integrand size = 80, antiderivative size = 20 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (1+x+\frac {x-\frac {390625}{\log (x)}}{\log (x \log (3))}\right ) \]
\[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx \]
Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3] ] + x*Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*L og[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]
Integrate[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3] ] + x*Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*L og[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2), 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.
| \(\displaystyle \int \frac {x \log ^2(x \log (3)) \log ^2(x)-x \log ^2(x)+\left (x \log ^2(x)+390625\right ) \log (x \log (3))+390625 \log (x)}{\left (x^2+x\right ) \log ^2(x) \log ^2(x \log (3))+\left (x^2 \log ^2(x)-390625 x \log (x)\right ) \log (x \log (3))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-x \log ^2(x \log (3)) \log ^2(x)+x \log ^2(x)-\left (x \log ^2(x)+390625\right ) \log (x \log (3))-390625 \log (x)}{x \log (x) \log (x \log (3)) (-x \log (x)-x \log (x \log (3)) \log (x)-\log (x \log (3)) \log (x)+390625)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{x+1}+\frac {390625 x+x \log ^2(x)+390625 x \log (x)+390625}{x (x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625) \log (x)}+\frac {(x+1) \log (x)}{x (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}-\frac {1}{x \log (x \log (3))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 390625 \int \frac {1}{(x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+390625 \int \frac {1}{x \log (x) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\int \frac {\log (x)}{x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625}dx+\int \frac {\log (x)}{x (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\int \frac {\log (x)}{(x+1) (x \log (x)+x \log (x \log (3)) \log (x)+\log (x \log (3)) \log (x)-390625)}dx+\log (x+1)-\log (\log (x \log (3)))\) |
Int[(390625*Log[x] - x*Log[x]^2 + (390625 + x*Log[x]^2)*Log[x*Log[3]] + x* Log[x]^2*Log[x*Log[3]]^2)/((-390625*x*Log[x] + x^2*Log[x]^2)*Log[x*Log[3]] + (x + x^2)*Log[x]^2*Log[x*Log[3]]^2),x]
3.18.84.3.1 Defintions of rubi rules used
Time = 1.71 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95
| method | result | size |
| parallelrisch | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \ln \left (3\right )\right )\right )+\ln \left (\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right ) x +\ln \left (x \right ) \ln \left (x \ln \left (3\right )\right )+x \ln \left (x \right )-390625\right )\) | \(39\) |
| default | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (\ln \left (3\right )\right )+\ln \left (x \right )\right )+\ln \left (x \ln \left (x \right ) \ln \left (\ln \left (3\right )\right )+x \ln \left (x \right )^{2}+\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}+x \ln \left (x \right )-390625\right )\) | \(46\) |
| risch | \(\ln \left (1+x \right )-\ln \left (\ln \left (\ln \left (3\right )\right ) \ln \left (x \right )+\ln \left (x \right )^{2}\right )+\ln \left (\ln \left (x \right )^{2}+\frac {\left (2 \ln \left (\ln \left (3\right )\right ) x +2 \ln \left (\ln \left (3\right )\right )+2 x \right ) \ln \left (x \right )}{2+2 x}-\frac {390625}{1+x}\right )\) | \(57\) |
int((x*ln(x)^2*ln(x*ln(3))^2+(x*ln(x)^2+390625)*ln(x*ln(3))-x*ln(x)^2+3906 25*ln(x))/((x^2+x)*ln(x)^2*ln(x*ln(3))^2+(x^2*ln(x)^2-390625*x*ln(x))*ln(x *ln(3))),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x + 1\right )} \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 0625*x*log(x))*log(x*log(3))),x, algorithm=\
log(x + 1) + log(((x + 1)*log(x)^2 + (x + 1)*log(x)*log(log(3)) + x*log(x) - 390625)/(x + 1)) - log(log(x) + log(log(3))) - log(log(x))
Exception generated. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\text {Exception raised: PolynomialError} \]
integrate((x*ln(x)**2*ln(x*ln(3))**2+(x*ln(x)**2+390625)*ln(x*ln(3))-x*ln( x)**2+390625*ln(x))/((x**2+x)*ln(x)**2*ln(x*ln(3))**2+(x**2*ln(x)**2-39062 5*x*ln(x))*ln(x*ln(3))),x)
Exception raised: PolynomialError >> 1/(x**6 + 4*x**5 + 6*x**4 + 4*x**3 + x**2) contains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (20) = 40\).
Time = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.50 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x + 1\right ) + \log \left (\frac {{\left (x + 1\right )} \log \left (x\right )^{2} + {\left (x {\left (\log \left (\log \left (3\right )\right ) + 1\right )} + \log \left (\log \left (3\right )\right )\right )} \log \left (x\right ) - 390625}{x + 1}\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 0625*x*log(x))*log(x*log(3))),x, algorithm=\
log(x + 1) + log(((x + 1)*log(x)^2 + (x*(log(log(3)) + 1) + log(log(3)))*l og(x) - 390625)/(x + 1)) - log(log(x) + log(log(3))) - log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.25 \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\log \left (x \log \left (x\right )^{2} + x \log \left (x\right ) \log \left (\log \left (3\right )\right ) + x \log \left (x\right ) + \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\log \left (3\right )\right ) - 390625\right ) - \log \left (\log \left (x\right ) + \log \left (\log \left (3\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((x*log(x)^2*log(x*log(3))^2+(x*log(x)^2+390625)*log(x*log(3))-x* log(x)^2+390625*log(x))/((x^2+x)*log(x)^2*log(x*log(3))^2+(x^2*log(x)^2-39 0625*x*log(x))*log(x*log(3))),x, algorithm=\
log(x*log(x)^2 + x*log(x)*log(log(3)) + x*log(x) + log(x)^2 + log(x)*log(l og(3)) - 390625) - log(log(x) + log(log(3))) - log(log(x))
Timed out. \[ \int \frac {390625 \log (x)-x \log ^2(x)+\left (390625+x \log ^2(x)\right ) \log (x \log (3))+x \log ^2(x) \log ^2(x \log (3))}{\left (-390625 x \log (x)+x^2 \log ^2(x)\right ) \log (x \log (3))+\left (x+x^2\right ) \log ^2(x) \log ^2(x \log (3))} \, dx=\int \frac {390625\,\ln \left (x\right )-x\,{\ln \left (x\right )}^2+\ln \left (x\,\ln \left (3\right )\right )\,\left (x\,{\ln \left (x\right )}^2+390625\right )+x\,{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2}{\ln \left (x\,\ln \left (3\right )\right )\,\left (x^2\,{\ln \left (x\right )}^2-390625\,x\,\ln \left (x\right )\right )+{\ln \left (x\,\ln \left (3\right )\right )}^2\,{\ln \left (x\right )}^2\,\left (x^2+x\right )} \,d x \]
int((390625*log(x) - x*log(x)^2 + log(x*log(3))*(x*log(x)^2 + 390625) + x* log(x*log(3))^2*log(x)^2)/(log(x*log(3))*(x^2*log(x)^2 - 390625*x*log(x)) + log(x*log(3))^2*log(x)^2*(x + x^2)),x)