Integrand size = 92, antiderivative size = 23 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\left (6-e^{\log ^2(\log (x))}+\frac {11 x}{3}\right )^2 \log ^2(x) \] Output:
ln(x)^2*(11/3*x+6-exp(ln(ln(x))^2))^2
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\frac {1}{9} \left (18-3 e^{\log ^2(\log (x))}+11 x\right )^2 \log ^2(x) \] Input:
Integrate[((648 + 792*x + 242*x^2)*Log[x] + (396*x + 242*x^2)*Log[x]^2 + E ^(2*Log[Log[x]]^2)*(18*Log[x] + 36*Log[x]*Log[Log[x]]) + E^Log[Log[x]]^2*( (-216 - 132*x)*Log[x] - 66*x*Log[x]^2 + (-216 - 132*x)*Log[x]*Log[Log[x]]) )/(9*x),x]
Output:
((18 - 3*E^Log[Log[x]]^2 + 11*x)^2*Log[x]^2)/9
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(23)=46\).
Time = 1.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 27, 2010, 2009}
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.
\(\displaystyle \int \frac {\left (242 x^2+396 x\right ) \log ^2(x)+\left (242 x^2+792 x+648\right ) \log (x)+e^{2 \log ^2(\log (x))} (36 \log (\log (x)) \log (x)+18 \log (x))+e^{\log ^2(\log (x))} \left (-66 x \log ^2(x)+(-132 x-216) \log (x)+(-132 x-216) \log (\log (x)) \log (x)\right )}{9 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {2 \left (11 \left (11 x^2+18 x\right ) \log ^2(x)+\left (121 x^2+396 x+324\right ) \log (x)+9 e^{2 \log ^2(\log (x))} (2 \log (\log (x)) \log (x)+\log (x))-3 e^{\log ^2(\log (x))} \left (11 x \log ^2(x)+2 (11 x+18) \log (x)+2 (11 x+18) \log (\log (x)) \log (x)\right )\right )}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{9} \int \frac {11 \left (11 x^2+18 x\right ) \log ^2(x)+\left (121 x^2+396 x+324\right ) \log (x)+9 e^{2 \log ^2(\log (x))} (2 \log (\log (x)) \log (x)+\log (x))-3 e^{\log ^2(\log (x))} \left (11 x \log ^2(x)+2 (11 x+18) \log (x)+2 (11 x+18) \log (\log (x)) \log (x)\right )}{x}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {2}{9} \int \left (\frac {(11 x+18) \log (x) (11 \log (x) x+11 x+18)}{x}+\frac {9 e^{2 \log ^2(\log (x))} \log (x) (2 \log (\log (x))+1)}{x}-\frac {3 e^{\log ^2(\log (x))} \log (x) (11 \log (x) x+22 \log (\log (x)) x+22 x+36 \log (\log (x))+36)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{9} \left (\frac {121}{2} x^2 \log ^2(x)+\frac {9}{2} e^{2 \log ^2(\log (x))} \log ^2(x)+198 x \log ^2(x)-\frac {3 e^{\log ^2(\log (x))} (11 x \log (\log (x))+18 \log (\log (x))) \log ^2(x)}{\log (\log (x))}+162 \log ^2(x)\right )\) |
Input:
Int[((648 + 792*x + 242*x^2)*Log[x] + (396*x + 242*x^2)*Log[x]^2 + E^(2*Lo g[Log[x]]^2)*(18*Log[x] + 36*Log[x]*Log[Log[x]]) + E^Log[Log[x]]^2*((-216 - 132*x)*Log[x] - 66*x*Log[x]^2 + (-216 - 132*x)*Log[x]*Log[Log[x]]))/(9*x ),x]
Output:
(2*(162*Log[x]^2 + (9*E^(2*Log[Log[x]]^2)*Log[x]^2)/2 + 198*x*Log[x]^2 + ( 121*x^2*Log[x]^2)/2 - (3*E^Log[Log[x]]^2*Log[x]^2*(18*Log[Log[x]] + 11*x*L og[Log[x]]))/Log[Log[x]]))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(20)=40\).
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09
method | result | size |
risch | \(\frac {\left (121 x^{2}+396 x +324\right ) \ln \left (x \right )^{2}}{9}+\ln \left (x \right )^{2} {\mathrm e}^{2 \ln \left (\ln \left (x \right )\right )^{2}}-\frac {2 \ln \left (x \right )^{2} \left (11 x +18\right ) {\mathrm e}^{\ln \left (\ln \left (x \right )\right )^{2}}}{3}\) | \(48\) |
parallelrisch | \(\frac {121 x^{2} \ln \left (x \right )^{2}}{9}+44 x \ln \left (x \right )^{2}+36 \ln \left (x \right )^{2}+\ln \left (x \right )^{2} {\mathrm e}^{2 \ln \left (\ln \left (x \right )\right )^{2}}-\frac {22 \,{\mathrm e}^{\ln \left (\ln \left (x \right )\right )^{2}} \ln \left (x \right )^{2} x}{3}-12 \ln \left (x \right )^{2} {\mathrm e}^{\ln \left (\ln \left (x \right )\right )^{2}}\) | \(62\) |
Input:
int(1/9*((36*ln(x)*ln(ln(x))+18*ln(x))*exp(ln(ln(x))^2)^2+((-132*x-216)*ln (x)*ln(ln(x))-66*x*ln(x)^2+(-132*x-216)*ln(x))*exp(ln(ln(x))^2)+(242*x^2+3 96*x)*ln(x)^2+(242*x^2+792*x+648)*ln(x))/x,x,method=_RETURNVERBOSE)
Output:
1/9*(121*x^2+396*x+324)*ln(x)^2+ln(x)^2*exp(ln(ln(x))^2)^2-2/3*ln(x)^2*(11 *x+18)*exp(ln(ln(x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=-\frac {2}{3} \, {\left (11 \, x + 18\right )} e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} \log \left (x\right )^{2} + \frac {1}{9} \, {\left (121 \, x^{2} + 396 \, x + 324\right )} \log \left (x\right )^{2} + e^{\left (2 \, \log \left (\log \left (x\right )\right )^{2}\right )} \log \left (x\right )^{2} \] Input:
integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-1 32*x-216)*log(x)*log(log(x))-66*x*log(x)^2+(-132*x-216)*log(x))*exp(log(lo g(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algorit hm="fricas")
Output:
-2/3*(11*x + 18)*e^(log(log(x))^2)*log(x)^2 + 1/9*(121*x^2 + 396*x + 324)* log(x)^2 + e^(2*log(log(x))^2)*log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 3.94 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\frac {\left (- 22 x \log {\left (x \right )}^{2} - 36 \log {\left (x \right )}^{2}\right ) e^{\log {\left (\log {\left (x \right )} \right )}^{2}}}{3} + \left (\frac {121 x^{2}}{9} + 44 x + 36\right ) \log {\left (x \right )}^{2} + e^{2 \log {\left (\log {\left (x \right )} \right )}^{2}} \log {\left (x \right )}^{2} \] Input:
integrate(1/9*((36*ln(x)*ln(ln(x))+18*ln(x))*exp(ln(ln(x))**2)**2+((-132*x -216)*ln(x)*ln(ln(x))-66*x*ln(x)**2+(-132*x-216)*ln(x))*exp(ln(ln(x))**2)+ (242*x**2+396*x)*ln(x)**2+(242*x**2+792*x+648)*ln(x))/x,x)
Output:
(-22*x*log(x)**2 - 36*log(x)**2)*exp(log(log(x))**2)/3 + (121*x**2/9 + 44* x + 36)*log(x)**2 + exp(2*log(log(x))**2)*log(x)**2
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=-\frac {2}{3} \, {\left (11 \, x + 18\right )} e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} \log \left (x\right )^{2} + \frac {121}{18} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + \frac {121}{9} \, x^{2} \log \left (x\right ) + e^{\left (2 \, \log \left (\log \left (x\right )\right )^{2}\right )} \log \left (x\right )^{2} + 44 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - \frac {121}{18} \, x^{2} + 88 \, x \log \left (x\right ) + 36 \, \log \left (x\right )^{2} - 88 \, x \] Input:
integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-1 32*x-216)*log(x)*log(log(x))-66*x*log(x)^2+(-132*x-216)*log(x))*exp(log(lo g(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algorit hm="maxima")
Output:
-2/3*(11*x + 18)*e^(log(log(x))^2)*log(x)^2 + 121/18*(2*log(x)^2 - 2*log(x ) + 1)*x^2 + 121/9*x^2*log(x) + e^(2*log(log(x))^2)*log(x)^2 + 44*(log(x)^ 2 - 2*log(x) + 2)*x - 121/18*x^2 + 88*x*log(x) + 36*log(x)^2 - 88*x
\[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\int { \frac {2 \, {\left (11 \, {\left (11 \, x^{2} + 18 \, x\right )} \log \left (x\right )^{2} + 9 \, {\left (2 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + \log \left (x\right )\right )} e^{\left (2 \, \log \left (\log \left (x\right )\right )^{2}\right )} - 3 \, {\left (11 \, x \log \left (x\right )^{2} + 2 \, {\left (11 \, x + 18\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, {\left (11 \, x + 18\right )} \log \left (x\right )\right )} e^{\left (\log \left (\log \left (x\right )\right )^{2}\right )} + {\left (121 \, x^{2} + 396 \, x + 324\right )} \log \left (x\right )\right )}}{9 \, x} \,d x } \] Input:
integrate(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-1 32*x-216)*log(x)*log(log(x))-66*x*log(x)^2+(-132*x-216)*log(x))*exp(log(lo g(x))^2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x, algorit hm="giac")
Output:
integrate(2/9*(11*(11*x^2 + 18*x)*log(x)^2 + 9*(2*log(x)*log(log(x)) + log (x))*e^(2*log(log(x))^2) - 3*(11*x*log(x)^2 + 2*(11*x + 18)*log(x)*log(log (x)) + 2*(11*x + 18)*log(x))*e^(log(log(x))^2) + (121*x^2 + 396*x + 324)*l og(x))/x, x)
Time = 2.71 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\frac {{\ln \left (x\right )}^2\,{\left (11\,x-3\,{\mathrm {e}}^{{\ln \left (\ln \left (x\right )\right )}^2}+18\right )}^2}{9} \] Input:
int(((exp(2*log(log(x))^2)*(18*log(x) + 36*log(log(x))*log(x)))/9 + (log(x )^2*(396*x + 242*x^2))/9 - (exp(log(log(x))^2)*(66*x*log(x)^2 + log(x)*(13 2*x + 216) + log(log(x))*log(x)*(132*x + 216)))/9 + (log(x)*(792*x + 242*x ^2 + 648))/9)/x,x)
Output:
(log(x)^2*(11*x - 3*exp(log(log(x))^2) + 18)^2)/9
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {\left (648+792 x+242 x^2\right ) \log (x)+\left (396 x+242 x^2\right ) \log ^2(x)+e^{2 \log ^2(\log (x))} (18 \log (x)+36 \log (x) \log (\log (x)))+e^{\log ^2(\log (x))} \left ((-216-132 x) \log (x)-66 x \log ^2(x)+(-216-132 x) \log (x) \log (\log (x))\right )}{9 x} \, dx=\frac {\mathrm {log}\left (x \right )^{2} \left (9 e^{2 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}}-66 e^{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}} x -108 e^{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}}+121 x^{2}+396 x +324\right )}{9} \] Input:
int(1/9*((36*log(x)*log(log(x))+18*log(x))*exp(log(log(x))^2)^2+((-132*x-2 16)*log(x)*log(log(x))-66*x*log(x)^2+(-132*x-216)*log(x))*exp(log(log(x))^ 2)+(242*x^2+396*x)*log(x)^2+(242*x^2+792*x+648)*log(x))/x,x)
Output:
(log(x)**2*(9*e**(2*log(log(x))**2) - 66*e**(log(log(x))**2)*x - 108*e**(l og(log(x))**2) + 121*x**2 + 396*x + 324))/9