Integrand size = 75, antiderivative size = 23 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\frac {2 \left (\frac {13}{3}-x+(5+x) \log (x \log (x))\right )^2}{x} \] Output:
2*(ln(x*ln(x))*(5+x)+13/3-x)^2/x
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\frac {2 \left (169+9 x^2-12 x \log (x)-12 x \log (\log (x))+6 \left (65-3 x^2\right ) \log (x \log (x))+9 (5+x)^2 \log ^2(x \log (x))\right )}{9 x} \] Input:
Integrate[(780 - 24*x - 36*x^2 + (442 - 24*x - 18*x^2)*Log[x] + (900 + 360 *x + 36*x^2 + (120 + 360*x)*Log[x])*Log[x*Log[x]] + (-450 + 18*x^2)*Log[x] *Log[x*Log[x]]^2)/(9*x^2*Log[x]),x]
Output:
(2*(169 + 9*x^2 - 12*x*Log[x] - 12*x*Log[Log[x]] + 6*(65 - 3*x^2)*Log[x*Lo g[x]] + 9*(5 + x)^2*Log[x*Log[x]]^2))/(9*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.
| \(\displaystyle \int \frac {-36 x^2+\left (18 x^2-450\right ) \log (x) \log ^2(x \log (x))+\left (-18 x^2-24 x+442\right ) \log (x)+\left (36 x^2+360 x+(360 x+120) \log (x)+900\right ) \log (x \log (x))-24 x+780}{9 x^2 \log (x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {2 \left (-18 x^2-12 x-9 \left (25-x^2\right ) \log (x) \log ^2(x \log (x))+\left (-9 x^2-12 x+221\right ) \log (x)+6 \left (3 x^2+30 x+10 (3 x+1) \log (x)+75\right ) \log (x \log (x))+390\right )}{x^2 \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int \frac {-18 x^2-12 x-9 \left (25-x^2\right ) \log (x) \log ^2(x \log (x))+\left (-9 x^2-12 x+221\right ) \log (x)+6 \left (3 x^2+30 x+10 (3 x+1) \log (x)+75\right ) \log (x \log (x))+390}{x^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{9} \int \frac {(-3 x+3 (x+5) \log (x \log (x))+13) (6 (x+5)+\log (x) (3 x+3 (x-5) \log (x \log (x))+17))}{x^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {9 (x-5) (x+5) \log ^2(x \log (x))}{x^2}+\frac {6 \left (3 x^2+30 \log (x) x+30 x+10 \log (x)+75\right ) \log (x \log (x))}{x^2 \log (x)}-\frac {(3 x-13) (3 \log (x) x+6 x+17 \log (x)+30)}{x^2 \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{9} \left (-18 \int \frac {\operatorname {LogIntegral}(x)}{x \log (x)}dx-225 \int \frac {\log ^2(x \log (x))}{x^2}dx-6 \int \frac {(x+5) (3 x-13)}{x^2 \log (x)}dx+450 \int \frac {\log (x \log (x))}{x^2 \log (x)}dx+9 \int \log ^2(x \log (x))dx+180 \int \frac {\log (x \log (x))}{x}dx+180 \int \frac {\log (x \log (x))}{x \log (x)}dx-60 \log (x) \operatorname {ExpIntegralEi}(-\log (x))+60 (\log (x)+1) \operatorname {ExpIntegralEi}(-\log (x))-18 \operatorname {LogIntegral}(x) \log (x)+18 \operatorname {LogIntegral}(x) \log (x \log (x))+9 x-\frac {281}{x}-12 \log (x)-\frac {60 \log (x \log (x))}{x}\right )\) |
Input:
Int[(780 - 24*x - 36*x^2 + (442 - 24*x - 18*x^2)*Log[x] + (900 + 360*x + 3 6*x^2 + (120 + 360*x)*Log[x])*Log[x*Log[x]] + (-450 + 18*x^2)*Log[x]*Log[x *Log[x]]^2)/(9*x^2*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(21)=42\).
Time = 0.86 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13
| method | result | size |
| parallelrisch | \(\frac {18 x^{2} \ln \left (x \ln \left (x \right )\right )^{2}+338-36 x^{2} \ln \left (x \ln \left (x \right )\right )+180 \ln \left (x \ln \left (x \right )\right )^{2} x -24 x \ln \left (\ln \left (x \right )\right )+18 x^{2}-24 x \ln \left (x \right )+450 \ln \left (x \ln \left (x \right )\right )^{2}+780 \ln \left (x \ln \left (x \right )\right )}{9 x}\) | \(72\) |
| risch | \(\text {Expression too large to display}\) | \(1198\) |
Input:
int(1/9*((18*x^2-450)*ln(x)*ln(x*ln(x))^2+((360*x+120)*ln(x)+36*x^2+360*x+ 900)*ln(x*ln(x))+(-18*x^2-24*x+442)*ln(x)-36*x^2-24*x+780)/x^2/ln(x),x,met hod=_RETURNVERBOSE)
Output:
1/9/x*(18*x^2*ln(x*ln(x))^2+338-36*x^2*ln(x*ln(x))+180*ln(x*ln(x))^2*x-24* x*ln(ln(x))+18*x^2-24*x*ln(x)+450*ln(x*ln(x))^2+780*ln(x*ln(x)))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\frac {2 \, {\left (9 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (x \log \left (x\right )\right )^{2} + 9 \, x^{2} - 6 \, {\left (3 \, x^{2} + 2 \, x - 65\right )} \log \left (x \log \left (x\right )\right ) + 169\right )}}{9 \, x} \] Input:
integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36* x^2+360*x+900)*log(x*log(x))+(-18*x^2-24*x+442)*log(x)-36*x^2-24*x+780)/x^ 2/log(x),x, algorithm="fricas")
Output:
2/9*(9*(x^2 + 10*x + 25)*log(x*log(x))^2 + 9*x^2 - 6*(3*x^2 + 2*x - 65)*lo g(x*log(x)) + 169)/x
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.61 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=2 x - \frac {8 \log {\left (x \right )}}{3} - \frac {8 \log {\left (\log {\left (x \right )} \right )}}{3} + \frac {\left (260 - 12 x^{2}\right ) \log {\left (x \log {\left (x \right )} \right )}}{3 x} + \frac {\left (2 x^{2} + 20 x + 50\right ) \log {\left (x \log {\left (x \right )} \right )}^{2}}{x} + \frac {338}{9 x} \] Input:
integrate(1/9*((18*x**2-450)*ln(x)*ln(x*ln(x))**2+((360*x+120)*ln(x)+36*x* *2+360*x+900)*ln(x*ln(x))+(-18*x**2-24*x+442)*ln(x)-36*x**2-24*x+780)/x**2 /ln(x),x)
Output:
2*x - 8*log(x)/3 - 8*log(log(x))/3 + (260 - 12*x**2)*log(x*log(x))/(3*x) + (2*x**2 + 20*x + 50)*log(x*log(x))**2/x + 338/(9*x)
\[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\int { \frac {2 \, {\left (9 \, {\left (x^{2} - 25\right )} \log \left (x \log \left (x\right )\right )^{2} \log \left (x\right ) - 18 \, x^{2} + 6 \, {\left (3 \, x^{2} + 10 \, {\left (3 \, x + 1\right )} \log \left (x\right ) + 30 \, x + 75\right )} \log \left (x \log \left (x\right )\right ) - {\left (9 \, x^{2} + 12 \, x - 221\right )} \log \left (x\right ) - 12 \, x + 390\right )}}{9 \, x^{2} \log \left (x\right )} \,d x } \] Input:
integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36* x^2+360*x+900)*log(x*log(x))+(-18*x^2-24*x+442)*log(x)-36*x^2-24*x+780)/x^ 2/log(x),x, algorithm="maxima")
Output:
-2*x + 2/3*(3*(x^2 + 10*x + 25)*log(x)^2 + 3*(x^2 + 10*x + 25)*log(log(x)) ^2 + 6*x^2 - 2*(3*x^2 - 65)*log(x) - 2*(3*x^2 - 3*(x^2 + 10*x + 25)*log(x) - 65)*log(log(x)) + 130)/x - 442/9/x + 260/3*Ei(-log(x)) - 4*Ei(log(x)) + 2/9*integrate(6*(3*x^2 - 65)/(x^2*log(x)), x) - 8/3*log(x) - 8/3*log(log( x))
\[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\int { \frac {2 \, {\left (9 \, {\left (x^{2} - 25\right )} \log \left (x \log \left (x\right )\right )^{2} \log \left (x\right ) - 18 \, x^{2} + 6 \, {\left (3 \, x^{2} + 10 \, {\left (3 \, x + 1\right )} \log \left (x\right ) + 30 \, x + 75\right )} \log \left (x \log \left (x\right )\right ) - {\left (9 \, x^{2} + 12 \, x - 221\right )} \log \left (x\right ) - 12 \, x + 390\right )}}{9 \, x^{2} \log \left (x\right )} \,d x } \] Input:
integrate(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36* x^2+360*x+900)*log(x*log(x))+(-18*x^2-24*x+442)*log(x)-36*x^2-24*x+780)/x^ 2/log(x),x, algorithm="giac")
Output:
integrate(2/9*(9*(x^2 - 25)*log(x*log(x))^2*log(x) - 18*x^2 + 6*(3*x^2 + 1 0*(3*x + 1)*log(x) + 30*x + 75)*log(x*log(x)) - (9*x^2 + 12*x - 221)*log(x ) - 12*x + 390)/(x^2*log(x)), x)
Time = 1.67 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.00 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=2\,x+\frac {112\,\ln \left (\ln \left (x\right )\right )}{3}+\frac {112\,\ln \left (x\right )}{3}-\ln \left (x\,\ln \left (x\right )\right )\,\left (8\,x-\frac {4\,x^2-40\,x+\frac {260}{3}}{x}\right )+\frac {338}{9\,x}+{\ln \left (x\,\ln \left (x\right )\right )}^2\,\left (4\,x-\frac {2\,x^2-50}{x}+20\right ) \] Input:
int(-((8*x)/3 - (log(x*log(x))*(360*x + log(x)*(360*x + 120) + 36*x^2 + 90 0))/9 + (log(x)*(24*x + 18*x^2 - 442))/9 + 4*x^2 - (log(x*log(x))^2*log(x) *(18*x^2 - 450))/9 - 260/3)/(x^2*log(x)),x)
Output:
2*x + (112*log(log(x)))/3 + (112*log(x))/3 - log(x*log(x))*(8*x - (4*x^2 - 40*x + 260/3)/x) + 338/(9*x) + log(x*log(x))^2*(4*x - (2*x^2 - 50)/x + 20 )
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx=\frac {-\frac {8 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x}{3}+2 \mathrm {log}\left (\mathrm {log}\left (x \right ) x \right )^{2} x^{2}+20 \mathrm {log}\left (\mathrm {log}\left (x \right ) x \right )^{2} x +50 \mathrm {log}\left (\mathrm {log}\left (x \right ) x \right )^{2}-4 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x \right ) x^{2}+\frac {260 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x \right )}{3}-\frac {8 \,\mathrm {log}\left (x \right ) x}{3}+2 x^{2}+\frac {338}{9}}{x} \] Input:
int(1/9*((18*x^2-450)*log(x)*log(x*log(x))^2+((360*x+120)*log(x)+36*x^2+36 0*x+900)*log(x*log(x))+(-18*x^2-24*x+442)*log(x)-36*x^2-24*x+780)/x^2/log( x),x)
Output:
(2*( - 12*log(log(x))*x + 9*log(log(x)*x)**2*x**2 + 90*log(log(x)*x)**2*x + 225*log(log(x)*x)**2 - 18*log(log(x)*x)*x**2 + 390*log(log(x)*x) - 12*lo g(x)*x + 9*x**2 + 169))/(9*x)