Integrand size = 111, antiderivative size = 22 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-\log \left (2+x (x-\log (25))^2 (5-\log (\log (x)))\right ) \] Output:
-ln((x-2*ln(5))^2*x*(5-ln(ln(x)))+2)
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
Time = 0.63 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-\log \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x^3 \log (\log (x))+2 x^2 \log (25) \log (\log (x))-x \log ^2(25) \log (\log (x))\right ) \] Input:
Integrate[(-x^2 + 2*x*Log[25] - Log[25]^2 + (15*x^2 - 20*x*Log[25] + 5*Log [25]^2)*Log[x] + (-3*x^2 + 4*x*Log[25] - Log[25]^2)*Log[x]*Log[Log[x]])/(( -2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2)*Log[x] + (x^3 - 2*x^2*Log[25] + x*Log[25]^2)*Log[x]*Log[Log[x]]),x]
Output:
-Log[2 + 5*x^3 - 10*x^2*Log[25] + 5*x*Log[25]^2 - x^3*Log[Log[x]] + 2*x^2* Log[25]*Log[Log[x]] - x*Log[25]^2*Log[Log[x]]]
Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(22)=44\).
Time = 0.80 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {7292, 7235}
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.
\(\displaystyle \int \frac {-x^2+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))+2 x \log (25)-\log ^2(25)}{\left (-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)-2\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (\log (x)) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(x-\log (25)) (x-15 x \log (x)+3 x \log (x) \log (\log (x))+5 \log (25) \log (x)-\log (25) \log (x) \log (\log (x))-\log (25))}{\log (x) \left (5 x^3+x^3 (-\log (\log (x)))+2 x^2 \log (25) \log (\log (x))-10 x^2 \log (25)-x \log ^2(25) \log (\log (x))+5 x \log ^2(25)+2\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle -\log \left (5 x^3+x^3 (-\log (\log (x)))+2 x^2 \log (25) \log (\log (x))-10 x^2 \log (25)-x \log ^2(25) \log (\log (x))+5 x \log ^2(25)+2\right )\) |
Input:
Int[(-x^2 + 2*x*Log[25] - Log[25]^2 + (15*x^2 - 20*x*Log[25] + 5*Log[25]^2 )*Log[x] + (-3*x^2 + 4*x*Log[25] - Log[25]^2)*Log[x]*Log[Log[x]])/((-2 - 5 *x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2)*Log[x] + (x^3 - 2*x^2*Log[25] + x*L og[25]^2)*Log[x]*Log[Log[x]]),x]
Output:
-Log[2 + 5*x^3 - 10*x^2*Log[25] + 5*x*Log[25]^2 - x^3*Log[Log[x]] + 2*x^2* Log[25]*Log[Log[x]] - x*Log[25]^2*Log[Log[x]]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 1.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36
method | result | size |
parallelrisch | \(-\ln \left (4 \ln \left (5\right )^{2} \ln \left (\ln \left (x \right )\right ) x -4 \ln \left (5\right ) \ln \left (\ln \left (x \right )\right ) x^{2}+\ln \left (\ln \left (x \right )\right ) x^{3}-20 x \ln \left (5\right )^{2}+20 x^{2} \ln \left (5\right )-5 x^{3}-2\right )\) | \(52\) |
risch | \(-\ln \left (x \right )-2 \ln \left (x -2 \ln \left (5\right )\right )-\ln \left (\ln \left (\ln \left (x \right )\right )-\frac {20 x \ln \left (5\right )^{2}-20 x^{2} \ln \left (5\right )+5 x^{3}+2}{x \left (4 \ln \left (5\right )^{2}-4 x \ln \left (5\right )+x^{2}\right )}\right )\) | \(65\) |
Input:
int(((-4*ln(5)^2+8*x*ln(5)-3*x^2)*ln(x)*ln(ln(x))+(20*ln(5)^2-40*x*ln(5)+1 5*x^2)*ln(x)-4*ln(5)^2+4*x*ln(5)-x^2)/((4*x*ln(5)^2-4*x^2*ln(5)+x^3)*ln(x) *ln(ln(x))+(-20*x*ln(5)^2+20*x^2*ln(5)-5*x^3-2)*ln(x)),x,method=_RETURNVER BOSE)
Output:
-ln(4*ln(5)^2*ln(ln(x))*x-4*ln(5)*ln(ln(x))*x^2+ln(ln(x))*x^3-20*x*ln(5)^2 +20*x^2*ln(5)-5*x^3-2)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.77 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-2 \, \log \left (x - 2 \, \log \left (5\right )\right ) - \log \left (x\right ) - \log \left (-\frac {5 \, x^{3} - 20 \, x^{2} \log \left (5\right ) + 20 \, x \log \left (5\right )^{2} - {\left (x^{3} - 4 \, x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right )^{2}\right )} \log \left (\log \left (x\right )\right ) + 2}{x^{3} - 4 \, x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right )^{2}}\right ) \] Input:
integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2- 40*x*log(5)+15*x^2)*log(x)-4*log(5)^2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2 *log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*log (x)),x, algorithm="fricas")
Output:
-2*log(x - 2*log(5)) - log(x) - log(-(5*x^3 - 20*x^2*log(5) + 20*x*log(5)^ 2 - (x^3 - 4*x^2*log(5) + 4*x*log(5)^2)*log(log(x)) + 2)/(x^3 - 4*x^2*log( 5) + 4*x*log(5)^2))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=- \log {\left (x \right )} - 2 \log {\left (x - 2 \log {\left (5 \right )} \right )} - \log {\left (\log {\left (\log {\left (x \right )} \right )} + \frac {- 5 x^{3} + 20 x^{2} \log {\left (5 \right )} - 20 x \log {\left (5 \right )}^{2} - 2}{x^{3} - 4 x^{2} \log {\left (5 \right )} + 4 x \log {\left (5 \right )}^{2}} \right )} \] Input:
integrate(((-4*ln(5)**2+8*x*ln(5)-3*x**2)*ln(x)*ln(ln(x))+(20*ln(5)**2-40* x*ln(5)+15*x**2)*ln(x)-4*ln(5)**2+4*x*ln(5)-x**2)/((4*x*ln(5)**2-4*x**2*ln (5)+x**3)*ln(x)*ln(ln(x))+(-20*x*ln(5)**2+20*x**2*ln(5)-5*x**3-2)*ln(x)),x )
Output:
-log(x) - 2*log(x - 2*log(5)) - log(log(log(x)) + (-5*x**3 + 20*x**2*log(5 ) - 20*x*log(5)**2 - 2)/(x**3 - 4*x**2*log(5) + 4*x*log(5)**2))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (21) = 42\).
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.77 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-2 \, \log \left (x - 2 \, \log \left (5\right )\right ) - \log \left (x\right ) - \log \left (-\frac {5 \, x^{3} - 20 \, x^{2} \log \left (5\right ) + 20 \, x \log \left (5\right )^{2} - {\left (x^{3} - 4 \, x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right )^{2}\right )} \log \left (\log \left (x\right )\right ) + 2}{x^{3} - 4 \, x^{2} \log \left (5\right ) + 4 \, x \log \left (5\right )^{2}}\right ) \] Input:
integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2- 40*x*log(5)+15*x^2)*log(x)-4*log(5)^2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2 *log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*log (x)),x, algorithm="maxima")
Output:
-2*log(x - 2*log(5)) - log(x) - log(-(5*x^3 - 20*x^2*log(5) + 20*x*log(5)^ 2 - (x^3 - 4*x^2*log(5) + 4*x*log(5)^2)*log(log(x)) + 2)/(x^3 - 4*x^2*log( 5) + 4*x*log(5)^2))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-\log \left (x^{3} \log \left (\log \left (x\right )\right ) - 4 \, x^{2} \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 4 \, x \log \left (5\right )^{2} \log \left (\log \left (x\right )\right ) - 5 \, x^{3} + 20 \, x^{2} \log \left (5\right ) - 20 \, x \log \left (5\right )^{2} - 2\right ) \] Input:
integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2- 40*x*log(5)+15*x^2)*log(x)-4*log(5)^2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2 *log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*log (x)),x, algorithm="giac")
Output:
-log(x^3*log(log(x)) - 4*x^2*log(5)*log(log(x)) + 4*x*log(5)^2*log(log(x)) - 5*x^3 + 20*x^2*log(5) - 20*x*log(5)^2 - 2)
Timed out. \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=\int \frac {4\,{\ln \left (5\right )}^2-\ln \left (x\right )\,\left (15\,x^2-40\,\ln \left (5\right )\,x+20\,{\ln \left (5\right )}^2\right )-4\,x\,\ln \left (5\right )+x^2+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (3\,x^2-8\,\ln \left (5\right )\,x+4\,{\ln \left (5\right )}^2\right )}{\ln \left (x\right )\,\left (5\,x^3-20\,\ln \left (5\right )\,x^2+20\,{\ln \left (5\right )}^2\,x+2\right )-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (x^3-4\,\ln \left (5\right )\,x^2+4\,{\ln \left (5\right )}^2\,x\right )} \,d x \] Input:
int((4*log(5)^2 - log(x)*(20*log(5)^2 - 40*x*log(5) + 15*x^2) - 4*x*log(5) + x^2 + log(log(x))*log(x)*(4*log(5)^2 - 8*x*log(5) + 3*x^2))/(log(x)*(20 *x*log(5)^2 - 20*x^2*log(5) + 5*x^3 + 2) - log(log(x))*log(x)*(4*x*log(5)^ 2 - 4*x^2*log(5) + x^3)),x)
Output:
int((4*log(5)^2 - log(x)*(20*log(5)^2 - 40*x*log(5) + 15*x^2) - 4*x*log(5) + x^2 + log(log(x))*log(x)*(4*log(5)^2 - 8*x*log(5) + 3*x^2))/(log(x)*(20 *x*log(5)^2 - 20*x^2*log(5) + 5*x^3 + 2) - log(log(x))*log(x)*(4*x*log(5)^ 2 - 4*x^2*log(5) + x^3)), x)
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx=-\mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right )^{2} x -4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (5\right ) x^{2}+\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{3}-20 \mathrm {log}\left (5\right )^{2} x +20 \,\mathrm {log}\left (5\right ) x^{2}-5 x^{3}-2\right ) \] Input:
int(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2-40*x*l og(5)+15*x^2)*log(x)-4*log(5)^2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2*log(5 )+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*log(x)),x )
Output:
- log(4*log(log(x))*log(5)**2*x - 4*log(log(x))*log(5)*x**2 + log(log(x)) *x**3 - 20*log(5)**2*x + 20*log(5)*x**2 - 5*x**3 - 2)