Integrand size = 143, antiderivative size = 30 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=\frac {1}{4}-\log (x) \left (e^{3 (4+x) \left (1-x^2\right )}-\log (\log (x))\right )^2 \] Output:
1/4-ln(x)*(exp((4+x)*(-3*x^2+3))-ln(ln(x)))^2
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=-e^{-6 x^2 (4+x)} \log (x) \left (e^{3 (4+x)}-e^{3 x^2 (4+x)} \log (\log (x))\right )^2 \] Input:
Integrate[(-E^(24 + 6*x - 24*x^2 - 6*x^3) + 2*E^(12 + 3*x - 12*x^2 - 3*x^3 ) + E^(24 + 6*x - 24*x^2 - 6*x^3)*(-6*x + 48*x^2 + 18*x^3)*Log[x] + (-2 + 2*E^(12 + 3*x - 12*x^2 - 3*x^3) + E^(12 + 3*x - 12*x^2 - 3*x^3)*(6*x - 48* x^2 - 18*x^3)*Log[x])*Log[Log[x]] - Log[Log[x]]^2)/x,x]
Output:
-((Log[x]*(E^(3*(4 + x)) - E^(3*x^2*(4 + x))*Log[Log[x]])^2)/E^(6*x^2*(4 + x)))
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(30)=60\).
Time = 0.58 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {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.
\(\displaystyle \int \frac {-e^{-6 x^3-24 x^2+6 x+24}+2 e^{-3 x^3-12 x^2+3 x+12}+\left (2 e^{-3 x^3-12 x^2+3 x+12}+e^{-3 x^3-12 x^2+3 x+12} \left (-18 x^3-48 x^2+6 x\right ) \log (x)-2\right ) \log (\log (x))+e^{-6 x^3-24 x^2+6 x+24} \left (18 x^3+48 x^2-6 x\right ) \log (x)-\log ^2(\log (x))}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {e^{-6 x^3-24 x^2+6 x+24} \left (18 x^3 \log (x)+48 x^2 \log (x)-6 x \log (x)-1\right )}{x}-\frac {2 e^{-3 x^3-12 x^2+3 x+12} \left (9 x^3 \log (x) \log (\log (x))+24 x^2 \log (x) \log (\log (x))-3 x \log (x) \log (\log (x))-\log (\log (x))-1\right )}{x}-\frac {\log (\log (x)) (\log (\log (x))+2)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e^{-6 x^3-24 x^2+6 x+24} \left (-3 x^3 \log (x)-8 x^2 \log (x)+x \log (x)\right )}{x \left (-3 x^2-8 x+1\right )}+\frac {2 e^{-3 x^3-12 x^2+3 x+12} \left (-3 x^3 \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))+x \log (x) \log (\log (x))\right )}{x \left (-3 x^2-8 x+1\right )}-\log (x) \log ^2(\log (x))\) |
Input:
Int[(-E^(24 + 6*x - 24*x^2 - 6*x^3) + 2*E^(12 + 3*x - 12*x^2 - 3*x^3) + E^ (24 + 6*x - 24*x^2 - 6*x^3)*(-6*x + 48*x^2 + 18*x^3)*Log[x] + (-2 + 2*E^(1 2 + 3*x - 12*x^2 - 3*x^3) + E^(12 + 3*x - 12*x^2 - 3*x^3)*(6*x - 48*x^2 - 18*x^3)*Log[x])*Log[Log[x]] - Log[Log[x]]^2)/x,x]
Output:
-((E^(24 + 6*x - 24*x^2 - 6*x^3)*(x*Log[x] - 8*x^2*Log[x] - 3*x^3*Log[x])) /(x*(1 - 8*x - 3*x^2))) - Log[x]*Log[Log[x]]^2 + (2*E^(12 + 3*x - 12*x^2 - 3*x^3)*(x*Log[x]*Log[Log[x]] - 8*x^2*Log[x]*Log[Log[x]] - 3*x^3*Log[x]*Lo g[Log[x]]))/(x*(1 - 8*x - 3*x^2))
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]]
Time = 0.50 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53
method | result | size |
risch | \(-{\mathrm e}^{-6 \left (-1+x \right ) \left (4+x \right ) \left (1+x \right )} \ln \left (x \right )+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-3 \left (-1+x \right ) \left (4+x \right ) \left (1+x \right )}-\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(46\) |
parallelrisch | \(-{\mathrm e}^{-6 x^{3}-24 x^{2}+6 x +24} \ln \left (x \right )+2 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{-3 x^{3}-12 x^{2}+3 x +12}-\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}\) | \(56\) |
Input:
int((-ln(ln(x))^2+((-18*x^3-48*x^2+6*x)*exp(-3*x^3-12*x^2+3*x+12)*ln(x)+2* exp(-3*x^3-12*x^2+3*x+12)-2)*ln(ln(x))+(18*x^3+48*x^2-6*x)*exp(-3*x^3-12*x ^2+3*x+12)^2*ln(x)-exp(-3*x^3-12*x^2+3*x+12)^2+2*exp(-3*x^3-12*x^2+3*x+12) )/x,x,method=_RETURNVERBOSE)
Output:
-exp(-6*(-1+x)*(4+x)*(1+x))*ln(x)+2*ln(x)*ln(ln(x))*exp(-3*(-1+x)*(4+x)*(1 +x))-ln(x)*ln(ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=2 \, e^{\left (-3 \, x^{3} - 12 \, x^{2} + 3 \, x + 12\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - e^{\left (-6 \, x^{3} - 24 \, x^{2} + 6 \, x + 24\right )} \log \left (x\right ) \] Input:
integrate((-log(log(x))^2+((-18*x^3-48*x^2+6*x)*exp(-3*x^3-12*x^2+3*x+12)* log(x)+2*exp(-3*x^3-12*x^2+3*x+12)-2)*log(log(x))+(18*x^3+48*x^2-6*x)*exp( -3*x^3-12*x^2+3*x+12)^2*log(x)-exp(-3*x^3-12*x^2+3*x+12)^2+2*exp(-3*x^3-12 *x^2+3*x+12))/x,x, algorithm="fricas")
Output:
2*e^(-3*x^3 - 12*x^2 + 3*x + 12)*log(x)*log(log(x)) - log(x)*log(log(x))^2 - e^(-6*x^3 - 24*x^2 + 6*x + 24)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 17.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=- e^{- 6 x^{3} - 24 x^{2} + 6 x + 24} \log {\left (x \right )} + 2 e^{- 3 x^{3} - 12 x^{2} + 3 x + 12} \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}^{2} \] Input:
integrate((-ln(ln(x))**2+((-18*x**3-48*x**2+6*x)*exp(-3*x**3-12*x**2+3*x+1 2)*ln(x)+2*exp(-3*x**3-12*x**2+3*x+12)-2)*ln(ln(x))+(18*x**3+48*x**2-6*x)* exp(-3*x**3-12*x**2+3*x+12)**2*ln(x)-exp(-3*x**3-12*x**2+3*x+12)**2+2*exp( -3*x**3-12*x**2+3*x+12))/x,x)
Output:
-exp(-6*x**3 - 24*x**2 + 6*x + 24)*log(x) + 2*exp(-3*x**3 - 12*x**2 + 3*x + 12)*log(x)*log(log(x)) - log(x)*log(log(x))**2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=-{\left (e^{\left (24 \, x^{2}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 2 \, e^{\left (-3 \, x^{3} + 12 \, x^{2} + 3 \, x + 12\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + e^{\left (-6 \, x^{3} + 6 \, x + 24\right )} \log \left (x\right )\right )} e^{\left (-24 \, x^{2}\right )} \] Input:
integrate((-log(log(x))^2+((-18*x^3-48*x^2+6*x)*exp(-3*x^3-12*x^2+3*x+12)* log(x)+2*exp(-3*x^3-12*x^2+3*x+12)-2)*log(log(x))+(18*x^3+48*x^2-6*x)*exp( -3*x^3-12*x^2+3*x+12)^2*log(x)-exp(-3*x^3-12*x^2+3*x+12)^2+2*exp(-3*x^3-12 *x^2+3*x+12))/x,x, algorithm="maxima")
Output:
-(e^(24*x^2)*log(x)*log(log(x))^2 - 2*e^(-3*x^3 + 12*x^2 + 3*x + 12)*log(x )*log(log(x)) + e^(-6*x^3 + 6*x + 24)*log(x))*e^(-24*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=2 \, e^{\left (-3 \, x^{3} - 12 \, x^{2} + 3 \, x + 12\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - e^{\left (-6 \, x^{3} - 24 \, x^{2} + 6 \, x + 24\right )} \log \left (x\right ) \] Input:
integrate((-log(log(x))^2+((-18*x^3-48*x^2+6*x)*exp(-3*x^3-12*x^2+3*x+12)* log(x)+2*exp(-3*x^3-12*x^2+3*x+12)-2)*log(log(x))+(18*x^3+48*x^2-6*x)*exp( -3*x^3-12*x^2+3*x+12)^2*log(x)-exp(-3*x^3-12*x^2+3*x+12)^2+2*exp(-3*x^3-12 *x^2+3*x+12))/x,x, algorithm="giac")
Output:
2*e^(-3*x^3 - 12*x^2 + 3*x + 12)*log(x)*log(log(x)) - log(x)*log(log(x))^2 - e^(-6*x^3 - 24*x^2 + 6*x + 24)*log(x)
Timed out. \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=\int -\frac {{\ln \left (\ln \left (x\right )\right )}^2+\left ({\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}\,\ln \left (x\right )\,\left (18\,x^3+48\,x^2-6\,x\right )-2\,{\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}+2\right )\,\ln \left (\ln \left (x\right )\right )-2\,{\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}+{\mathrm {e}}^{-6\,x^3-24\,x^2+6\,x+24}-{\mathrm {e}}^{-6\,x^3-24\,x^2+6\,x+24}\,\ln \left (x\right )\,\left (18\,x^3+48\,x^2-6\,x\right )}{x} \,d x \] Input:
int(-(exp(6*x - 24*x^2 - 6*x^3 + 24) - 2*exp(3*x - 12*x^2 - 3*x^3 + 12) + log(log(x))*(exp(3*x - 12*x^2 - 3*x^3 + 12)*log(x)*(48*x^2 - 6*x + 18*x^3) - 2*exp(3*x - 12*x^2 - 3*x^3 + 12) + 2) + log(log(x))^2 - exp(6*x - 24*x^ 2 - 6*x^3 + 24)*log(x)*(48*x^2 - 6*x + 18*x^3))/x,x)
Output:
int(-(exp(6*x - 24*x^2 - 6*x^3 + 24) - 2*exp(3*x - 12*x^2 - 3*x^3 + 12) + log(log(x))*(exp(3*x - 12*x^2 - 3*x^3 + 12)*log(x)*(48*x^2 - 6*x + 18*x^3) - 2*exp(3*x - 12*x^2 - 3*x^3 + 12) + 2) + log(log(x))^2 - exp(6*x - 24*x^ 2 - 6*x^3 + 24)*log(x)*(48*x^2 - 6*x + 18*x^3))/x, x)
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {-e^{24+6 x-24 x^2-6 x^3}+2 e^{12+3 x-12 x^2-3 x^3}+e^{24+6 x-24 x^2-6 x^3} \left (-6 x+48 x^2+18 x^3\right ) \log (x)+\left (-2+2 e^{12+3 x-12 x^2-3 x^3}+e^{12+3 x-12 x^2-3 x^3} \left (6 x-48 x^2-18 x^3\right ) \log (x)\right ) \log (\log (x))-\log ^2(\log (x))}{x} \, dx=\frac {\mathrm {log}\left (x \right ) \left (-e^{6 x^{3}+24 x^{2}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}+2 e^{3 x^{3}+12 x^{2}+3 x} \mathrm {log}\left (\mathrm {log}\left (x \right )\right ) e^{12}-e^{6 x} e^{24}\right )}{e^{6 x^{3}+24 x^{2}}} \] Input:
int((-log(log(x))^2+((-18*x^3-48*x^2+6*x)*exp(-3*x^3-12*x^2+3*x+12)*log(x) +2*exp(-3*x^3-12*x^2+3*x+12)-2)*log(log(x))+(18*x^3+48*x^2-6*x)*exp(-3*x^3 -12*x^2+3*x+12)^2*log(x)-exp(-3*x^3-12*x^2+3*x+12)^2+2*exp(-3*x^3-12*x^2+3 *x+12))/x,x)
Output:
(log(x)*( - e**(6*x**3 + 24*x**2)*log(log(x))**2 + 2*e**(3*x**3 + 12*x**2 + 3*x)*log(log(x))*e**12 - e**(6*x)*e**24))/e**(6*x**3 + 24*x**2)