Integrand size = 81, antiderivative size = 17 \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=\log \left (-1+x+\log (x)+\frac {1}{\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(80\) vs. \(2(17)=34\).
Time = 0.61 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.71 \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=-\log \left (\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )\right )+\log \left (2-2 \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+2 x \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )-\log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+2 \left (\frac {1}{2} \log \left (\frac {3}{x^2}\right )+\log (x)\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )\right ) \]
Integrate[(4 + (1 + x)*Log[3/x^2]*Log[Log[3/x^2]^2]^2)/(x*Log[3/x^2]*Log[L og[3/x^2]^2] + ((-x + x^2)*Log[3/x^2] + x*Log[3/x^2]*Log[x])*Log[Log[3/x^2 ]^2]^2),x]
-Log[Log[Log[3/x^2]^2]] + Log[2 - 2*Log[Log[3/x^2]^2] + 2*x*Log[Log[3/x^2] ^2] - Log[3/x^2]*Log[Log[3/x^2]^2] + 2*(Log[3/x^2]/2 + Log[x])*Log[Log[3/x ^2]^2]]
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.
| \(\displaystyle \int \frac {(x+1) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )+4}{\left (\left (x^2-x\right ) \log \left (\frac {3}{x^2}\right )+x \log (x) \log \left (\frac {3}{x^2}\right )\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )+x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {(x+1) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )+4}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right ) \left (x \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\log (x) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )-\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+1) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )+4}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right ) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-x-1}{x (x+\log (x)-1) \left (x \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\log (x) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )-\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}+\frac {4}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )}-\frac {4 (x+\log (x)-1)}{x \log \left (\frac {3}{x^2}\right ) \left (x \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\log (x) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )-\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}+\frac {x+1}{x (x+\log (x)-1)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \int \frac {1}{\log \left (\frac {3}{x^2}\right ) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx+4 \int \frac {1}{x \log \left (\frac {3}{x^2}\right ) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx-4 \int \frac {\log (x)}{x \log \left (\frac {3}{x^2}\right ) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx-\int \frac {1}{(x+\log (x)-1) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx-\int \frac {1}{x (x+\log (x)-1) \left ((x+\log (x)-1) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+1\right )}dx-\log \left (\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )\right )+\log (-x-\log (x)+1)\) |
Int[(4 + (1 + x)*Log[3/x^2]*Log[Log[3/x^2]^2]^2)/(x*Log[3/x^2]*Log[Log[3/x ^2]^2] + ((-x + x^2)*Log[3/x^2] + x*Log[3/x^2]*Log[x])*Log[Log[3/x^2]^2]^2 ),x]
3.20.45.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(17)=34\).
Time = 23.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.00
| method | result | size |
| parallelrisch | \(-\ln \left (\ln \left (\ln \left (\frac {3}{x^{2}}\right )^{2}\right )\right )+\ln \left (\ln \left (\ln \left (\frac {3}{x^{2}}\right )^{2}\right ) \ln \left (x \right )+\ln \left (\ln \left (\frac {3}{x^{2}}\right )^{2}\right ) x -\ln \left (\ln \left (\frac {3}{x^{2}}\right )^{2}\right )+1\right )\) | \(51\) |
| risch | \(\text {Expression too large to display}\) | \(1753\) |
int(((1+x)*ln(3/x^2)*ln(ln(3/x^2)^2)^2+4)/((x*ln(3/x^2)*ln(x)+(x^2-x)*ln(3 /x^2))*ln(ln(3/x^2)^2)^2+x*ln(3/x^2)*ln(ln(3/x^2)^2)),x,method=_RETURNVERB OSE)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (17) = 34\).
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 4.41 \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=\log \left (-2 \, x - \log \left (3\right ) + \log \left (\frac {3}{x^{2}}\right ) + 2\right ) + \log \left (\frac {{\left (2 \, x + \log \left (3\right ) - \log \left (\frac {3}{x^{2}}\right ) - 2\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right ) + 2}{2 \, x + \log \left (3\right ) - \log \left (\frac {3}{x^{2}}\right ) - 2}\right ) - \log \left (\log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )\right ) \]
integrate(((1+x)*log(3/x^2)*log(log(3/x^2)^2)^2+4)/((x*log(3/x^2)*log(x)+( x^2-x)*log(3/x^2))*log(log(3/x^2)^2)^2+x*log(3/x^2)*log(log(3/x^2)^2)),x, algorithm=\
log(-2*x - log(3) + log(3/x^2) + 2) + log(((2*x + log(3) - log(3/x^2) - 2) *log(log(3/x^2)^2) + 2)/(2*x + log(3) - log(3/x^2) - 2)) - log(log(log(3/x ^2)^2))
Exception generated. \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((1+x)*ln(3/x**2)*ln(ln(3/x**2)**2)**2+4)/((x*ln(3/x**2)*ln(x)+( x**2-x)*ln(3/x**2))*ln(ln(3/x**2)**2)**2+x*ln(3/x**2)*ln(ln(3/x**2)**2)),x )
Exception raised: PolynomialError >> 1/(2*_t0**3*x + 4*_t0**2*x**2 - 4*_t0 **2*x - _t0**2*x*log(3) + 2*_t0*x**3 - 4*_t0*x**2 - 2*_t0*x**2*log(3) + 2* _t0*x + 2*_t0*x*log(3) - x**3*log(3) + 2*x**2*log(3) - x*log(3)) contains an element of t
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (17) = 34\).
Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.88 \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=\log \left (x + \log \left (x\right ) - 1\right ) + \log \left (\frac {2 \, {\left (x + \log \left (x\right ) - 1\right )} \log \left (-\log \left (3\right ) + 2 \, \log \left (x\right )\right ) + 1}{2 \, {\left (x + \log \left (x\right ) - 1\right )}}\right ) - \log \left (\log \left (-\log \left (3\right ) + 2 \, \log \left (x\right )\right )\right ) \]
integrate(((1+x)*log(3/x^2)*log(log(3/x^2)^2)^2+4)/((x*log(3/x^2)*log(x)+( x^2-x)*log(3/x^2))*log(log(3/x^2)^2)^2+x*log(3/x^2)*log(log(3/x^2)^2)),x, algorithm=\
log(x + log(x) - 1) + log(1/2*(2*(x + log(x) - 1)*log(-log(3) + 2*log(x)) + 1)/(x + log(x) - 1)) - log(log(-log(3) + 2*log(x)))
\[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=\int { \frac {{\left (x + 1\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )^{2} \log \left (\frac {3}{x^{2}}\right ) + 4}{{\left (x \log \left (x\right ) \log \left (\frac {3}{x^{2}}\right ) + {\left (x^{2} - x\right )} \log \left (\frac {3}{x^{2}}\right )\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )^{2} + x \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right ) \log \left (\frac {3}{x^{2}}\right )} \,d x } \]
integrate(((1+x)*log(3/x^2)*log(log(3/x^2)^2)^2+4)/((x*log(3/x^2)*log(x)+( x^2-x)*log(3/x^2))*log(log(3/x^2)^2)^2+x*log(3/x^2)*log(log(3/x^2)^2)),x, algorithm=\
integrate(((x + 1)*log(log(3/x^2)^2)^2*log(3/x^2) + 4)/((x*log(x)*log(3/x^ 2) + (x^2 - x)*log(3/x^2))*log(log(3/x^2)^2)^2 + x*log(log(3/x^2)^2)*log(3 /x^2)), x)
Timed out. \[ \int \frac {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx=-\int \frac {\ln \left (\frac {3}{x^2}\right )\,\left (x+1\right )\,{\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )}^2+4}{{\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )}^2\,\left (\ln \left (\frac {3}{x^2}\right )\,\left (x-x^2\right )-x\,\ln \left (\frac {3}{x^2}\right )\,\ln \left (x\right )\right )-x\,\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )\,\ln \left (\frac {3}{x^2}\right )} \,d x \]
int(-(log(log(3/x^2)^2)^2*log(3/x^2)*(x + 1) + 4)/(log(log(3/x^2)^2)^2*(lo g(3/x^2)*(x - x^2) - x*log(3/x^2)*log(x)) - x*log(log(3/x^2)^2)*log(3/x^2) ),x)