Integrand size = 137, antiderivative size = 26 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{(-5+x) \left (3-x+3 (3+x) \log ^2\left (\log \left (x^2\right )\right )\right )} \]
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{(-5+x) \left (3-x+3 (3+x) \log ^2\left (\log \left (x^2\right )\right )\right )} \]
Integrate[((-15 + x^2)*Log[x^2] + (180 + 24*x - 12*x^2)*Log[Log[x^2]] + (- 45 - 3*x^2)*Log[x^2]*Log[Log[x^2]]^2)/((225 - 240*x + 94*x^2 - 16*x^3 + x^ 4)*Log[x^2] + (1350 - 540*x - 96*x^2 + 60*x^3 - 6*x^4)*Log[x^2]*Log[Log[x^ 2]]^2 + (2025 + 540*x - 234*x^2 - 36*x^3 + 9*x^4)*Log[x^2]*Log[Log[x^2]]^4 ),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.
| \(\displaystyle \int \frac {\left (-3 x^2-45\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-12 x^2+24 x+180\right ) \log \left (\log \left (x^2\right )\right )+\left (x^2-15\right ) \log \left (x^2\right )}{\left (9 x^4-36 x^3-234 x^2+540 x+2025\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )+\left (-6 x^4+60 x^3-96 x^2-540 x+1350\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (x^4-16 x^3+94 x^2-240 x+225\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log \left (x^2\right ) \left (x^2-3 \left (x^2+15\right ) \log ^2\left (\log \left (x^2\right )\right )-15\right )-12 \left (x^2-2 x-15\right ) \log \left (\log \left (x^2\right )\right )}{(5-x)^2 \log \left (x^2\right ) \left (3 (x+3) \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-x^2-15}{(x-5)^2 (x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}-\frac {6 \left (2 x^2 \log \left (\log \left (x^2\right )\right )-x \log \left (x^2\right )+12 x \log \left (\log \left (x^2\right )\right )+18 \log \left (\log \left (x^2\right )\right )\right )}{(x-5) (x+3) \log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {15}{4} \int \frac {1}{(x-5) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx+\frac {9}{4} \int \frac {1}{(x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-12 \int \frac {\log \left (\log \left (x^2\right )\right )}{\log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-96 \int \frac {\log \left (\log \left (x^2\right )\right )}{(x-5) \log \left (x^2\right ) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )^2}dx-5 \int \frac {1}{(x-5)^2 \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx-\frac {5}{8} \int \frac {1}{(x-5) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx-\frac {3}{8} \int \frac {1}{(x+3) \left (3 x \log ^2\left (\log \left (x^2\right )\right )+9 \log ^2\left (\log \left (x^2\right )\right )-x+3\right )}dx\) |
Int[((-15 + x^2)*Log[x^2] + (180 + 24*x - 12*x^2)*Log[Log[x^2]] + (-45 - 3 *x^2)*Log[x^2]*Log[Log[x^2]]^2)/((225 - 240*x + 94*x^2 - 16*x^3 + x^4)*Log [x^2] + (1350 - 540*x - 96*x^2 + 60*x^3 - 6*x^4)*Log[x^2]*Log[Log[x^2]]^2 + (2025 + 540*x - 234*x^2 - 36*x^3 + 9*x^4)*Log[x^2]*Log[Log[x^2]]^4),x]
3.1.10.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77
| method | result | size |
| parallelrisch | \(\frac {x}{3 x^{2} {\ln \left (\ln \left (x^{2}\right )\right )}^{2}-6 x {\ln \left (\ln \left (x^{2}\right )\right )}^{2}-45 {\ln \left (\ln \left (x^{2}\right )\right )}^{2}-x^{2}+8 x -15}\) | \(46\) |
int(((-3*x^2-45)*ln(x^2)*ln(ln(x^2))^2+(-12*x^2+24*x+180)*ln(ln(x^2))+(x^2 -15)*ln(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*ln(x^2)*ln(ln(x^2))^4+(-6 *x^4+60*x^3-96*x^2-540*x+1350)*ln(x^2)*ln(ln(x^2))^2+(x^4-16*x^3+94*x^2-24 0*x+225)*ln(x^2)),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{3 \, {\left (x^{2} - 2 \, x - 15\right )} \log \left (\log \left (x^{2}\right )\right )^{2} - x^{2} + 8 \, x - 15} \]
integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log (x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{- x^{2} + 8 x + \left (3 x^{2} - 6 x - 45\right ) \log {\left (\log {\left (x^{2} \right )} \right )}^{2} - 15} \]
integrate(((-3*x**2-45)*ln(x**2)*ln(ln(x**2))**2+(-12*x**2+24*x+180)*ln(ln (x**2))+(x**2-15)*ln(x**2))/((9*x**4-36*x**3-234*x**2+540*x+2025)*ln(x**2) *ln(ln(x**2))**4+(-6*x**4+60*x**3-96*x**2-540*x+1350)*ln(x**2)*ln(ln(x**2) )**2+(x**4-16*x**3+94*x**2-240*x+225)*ln(x**2)),x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{{\left (3 \, \log \left (2\right )^{2} - 1\right )} x^{2} + 3 \, {\left (x^{2} - 2 \, x - 15\right )} \log \left (\log \left (x\right )\right )^{2} - 2 \, {\left (3 \, \log \left (2\right )^{2} - 4\right )} x - 45 \, \log \left (2\right )^{2} + 6 \, {\left (x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right ) - 15 \, \log \left (2\right )\right )} \log \left (\log \left (x\right )\right ) - 15} \]
integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log (x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm=\
x/((3*log(2)^2 - 1)*x^2 + 3*(x^2 - 2*x - 15)*log(log(x))^2 - 2*(3*log(2)^2 - 4)*x - 45*log(2)^2 + 6*(x^2*log(2) - 2*x*log(2) - 15*log(2))*log(log(x) ) - 15)
Time = 0.86 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\frac {x}{3 \, x^{2} \log \left (\log \left (x^{2}\right )\right )^{2} - 6 \, x \log \left (\log \left (x^{2}\right )\right )^{2} - x^{2} - 45 \, \log \left (\log \left (x^{2}\right )\right )^{2} + 8 \, x - 15} \]
integrate(((-3*x^2-45)*log(x^2)*log(log(x^2))^2+(-12*x^2+24*x+180)*log(log (x^2))+(x^2-15)*log(x^2))/((9*x^4-36*x^3-234*x^2+540*x+2025)*log(x^2)*log( log(x^2))^4+(-6*x^4+60*x^3-96*x^2-540*x+1350)*log(x^2)*log(log(x^2))^2+(x^ 4-16*x^3+94*x^2-240*x+225)*log(x^2)),x, algorithm=\
Timed out. \[ \int \frac {\left (-15+x^2\right ) \log \left (x^2\right )+\left (180+24 x-12 x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (-45-3 x^2\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{\left (225-240 x+94 x^2-16 x^3+x^4\right ) \log \left (x^2\right )+\left (1350-540 x-96 x^2+60 x^3-6 x^4\right ) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (2025+540 x-234 x^2-36 x^3+9 x^4\right ) \log \left (x^2\right ) \log ^4\left (\log \left (x^2\right )\right )} \, dx=\int \frac {-\ln \left (x^2\right )\,\left (3\,x^2+45\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2+\left (-12\,x^2+24\,x+180\right )\,\ln \left (\ln \left (x^2\right )\right )+\ln \left (x^2\right )\,\left (x^2-15\right )}{\ln \left (x^2\right )\,\left (9\,x^4-36\,x^3-234\,x^2+540\,x+2025\right )\,{\ln \left (\ln \left (x^2\right )\right )}^4-\ln \left (x^2\right )\,\left (6\,x^4-60\,x^3+96\,x^2+540\,x-1350\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2+\ln \left (x^2\right )\,\left (x^4-16\,x^3+94\,x^2-240\,x+225\right )} \,d x \]
int((log(log(x^2))*(24*x - 12*x^2 + 180) + log(x^2)*(x^2 - 15) - log(x^2)* log(log(x^2))^2*(3*x^2 + 45))/(log(x^2)*(94*x^2 - 240*x - 16*x^3 + x^4 + 2 25) - log(x^2)*log(log(x^2))^2*(540*x + 96*x^2 - 60*x^3 + 6*x^4 - 1350) + log(x^2)*log(log(x^2))^4*(540*x - 234*x^2 - 36*x^3 + 9*x^4 + 2025)),x)
int((log(log(x^2))*(24*x - 12*x^2 + 180) + log(x^2)*(x^2 - 15) - log(x^2)* log(log(x^2))^2*(3*x^2 + 45))/(log(x^2)*(94*x^2 - 240*x - 16*x^3 + x^4 + 2 25) - log(x^2)*log(log(x^2))^2*(540*x + 96*x^2 - 60*x^3 + 6*x^4 - 1350) + log(x^2)*log(log(x^2))^4*(540*x - 234*x^2 - 36*x^3 + 9*x^4 + 2025)), x)