Integrand size = 75, antiderivative size = 33 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=(3-x) x \log (x) \left (-e^x+x+\frac {1}{5} (3-x-x \log (x \log (x)))\right ) \] Output:
(3-x)*ln(x)*(4/5*x-exp(x)-1/5*ln(x*ln(x))*x+3/5)*x
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\frac {1}{5} (-3+x) x \log (x) \left (-3+5 e^x-4 x+x \log (x \log (x))\right ) \] Input:
Integrate[(9 + 6*x - 3*x^2 + E^x*(-15 + 5*x) + (9 + 15*x - 11*x^2 + E^x*(- 15 - 5*x + 5*x^2))*Log[x] + (-3*x + x^2 + (-6*x + 3*x^2)*Log[x])*Log[x*Log [x]])/5,x]
Output:
((-3 + x)*x*Log[x]*(-3 + 5*E^x - 4*x + x*Log[x*Log[x]]))/5
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.
\(\displaystyle \int \frac {1}{5} \left (-3 x^2+\left (-11 x^2+e^x \left (5 x^2-5 x-15\right )+15 x+9\right ) \log (x)+\left (x^2+\left (3 x^2-6 x\right ) \log (x)-3 x\right ) \log (x \log (x))+6 x+e^x (5 x-15)+9\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \left (-3 x^2+6 x-5 e^x (3-x)+\left (-11 x^2+15 x-5 e^x \left (-x^2+x+3\right )+9\right ) \log (x)-\left (-x^2+3 x+3 \left (2 x-x^2\right ) \log (x)\right ) \log (x \log (x))+9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (3 \int x^2 \log (x) \log (x \log (x))dx-6 \int x \log (x) \log (x \log (x))dx-\frac {3}{2} \log (x) \operatorname {ExpIntegralEi}(2 \log (x))+\frac {1}{3} \log (x) \operatorname {ExpIntegralEi}(3 \log (x))+\frac {3}{2} (\log (x)+1) \operatorname {ExpIntegralEi}(2 \log (x))-\frac {1}{3} (\log (x)+1) \operatorname {ExpIntegralEi}(3 \log (x))+\frac {x^3}{9}-\frac {11}{3} x^3 \log (x)+\frac {1}{3} x^3 \log (x \log (x))+5 e^x x^2 \log (x)+\frac {15}{2} x^2 \log (x)-\frac {3}{2} x^2 \log (x \log (x))-15 e^x x \log (x)+9 x \log (x)\right )\) |
Input:
Int[(9 + 6*x - 3*x^2 + E^x*(-15 + 5*x) + (9 + 15*x - 11*x^2 + E^x*(-15 - 5 *x + 5*x^2))*Log[x] + (-3*x + x^2 + (-6*x + 3*x^2)*Log[x])*Log[x*Log[x]])/ 5,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).
Time = 0.65 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82
method | result | size |
default | \(\frac {\ln \left (x \right ) x^{3} \ln \left (x \ln \left (x \right )\right )}{5}+\frac {9 x^{2} \ln \left (x \right )}{5}-\frac {4 x^{3} \ln \left (x \right )}{5}-\frac {3 \ln \left (x \right ) \ln \left (x \ln \left (x \right )\right ) x^{2}}{5}+\frac {9 x \ln \left (x \right )}{5}+x^{2} {\mathrm e}^{x} \ln \left (x \right )-3 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(60\) |
parallelrisch | \(\frac {\ln \left (x \right ) x^{3} \ln \left (x \ln \left (x \right )\right )}{5}+\frac {9 x^{2} \ln \left (x \right )}{5}-\frac {4 x^{3} \ln \left (x \right )}{5}-\frac {3 \ln \left (x \right ) \ln \left (x \ln \left (x \right )\right ) x^{2}}{5}+\frac {9 x \ln \left (x \right )}{5}+x^{2} {\mathrm e}^{x} \ln \left (x \right )-3 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(60\) |
risch | \(\frac {\ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) x^{3}}{5}-\frac {3 x^{2} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )}{5}+\frac {x^{3} \ln \left (x \right )^{2}}{5}-\frac {3 x^{2} \ln \left (x \right )^{2}}{5}-\frac {i \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right ) \ln \left (x \right )}{10}-\frac {3 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \ln \left (x \right )}{10}+\frac {3 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right ) \ln \left (x \right )}{10}+\frac {3 i \pi \,x^{2} \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{3} \ln \left (x \right )}{10}+\frac {i \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \ln \left (x \right )}{10}+\frac {i \pi \,x^{3} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \ln \left (x \right )}{10}-\frac {3 i \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \ln \left (x \right )}{10}-\frac {i \pi \,x^{3} \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{3} \ln \left (x \right )}{10}-\frac {4 x^{3} \ln \left (x \right )}{5}+\frac {9 x^{2} \ln \left (x \right )}{5}+\frac {9 x \ln \left (x \right )}{5}+x^{2} {\mathrm e}^{x} \ln \left (x \right )-3 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(258\) |
Input:
int(1/5*((3*x^2-6*x)*ln(x)+x^2-3*x)*ln(x*ln(x))+1/5*((5*x^2-5*x-15)*exp(x) -11*x^2+15*x+9)*ln(x)+1/5*(5*x-15)*exp(x)-3/5*x^2+6/5*x+9/5,x,method=_RETU RNVERBOSE)
Output:
1/5*ln(x)*x^3*ln(x*ln(x))+9/5*x^2*ln(x)-4/5*x^3*ln(x)-3/5*ln(x)*ln(x*ln(x) )*x^2+9/5*x*ln(x)+x^2*exp(x)*ln(x)-3*x*exp(x)*ln(x)
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\frac {1}{5} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x \log \left (x\right )\right ) \log \left (x\right ) - \frac {1}{5} \, {\left (4 \, x^{3} - 9 \, x^{2} - 5 \, {\left (x^{2} - 3 \, x\right )} e^{x} - 9 \, x\right )} \log \left (x\right ) \] Input:
integrate(1/5*((3*x^2-6*x)*log(x)+x^2-3*x)*log(x*log(x))+1/5*((5*x^2-5*x-1 5)*exp(x)-11*x^2+15*x+9)*log(x)+1/5*(5*x-15)*exp(x)-3/5*x^2+6/5*x+9/5,x, a lgorithm="fricas")
Output:
1/5*(x^3 - 3*x^2)*log(x*log(x))*log(x) - 1/5*(4*x^3 - 9*x^2 - 5*(x^2 - 3*x )*e^x - 9*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.40 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\left (x^{2} \log {\left (x \right )} - 3 x \log {\left (x \right )}\right ) e^{x} + \left (\frac {x^{3} \log {\left (x \right )}}{5} - \frac {3 x^{2} \log {\left (x \right )}}{5}\right ) \log {\left (x \log {\left (x \right )} \right )} + \left (- \frac {4 x^{3}}{5} + \frac {9 x^{2}}{5} + \frac {9 x}{5}\right ) \log {\left (x \right )} \] Input:
integrate(1/5*((3*x**2-6*x)*ln(x)+x**2-3*x)*ln(x*ln(x))+1/5*((5*x**2-5*x-1 5)*exp(x)-11*x**2+15*x+9)*ln(x)+1/5*(5*x-15)*exp(x)-3/5*x**2+6/5*x+9/5,x)
Output:
(x**2*log(x) - 3*x*log(x))*exp(x) + (x**3*log(x)/5 - 3*x**2*log(x)/5)*log( x*log(x)) + (-4*x**3/5 + 9*x**2/5 + 9*x/5)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\frac {1}{5} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x \log \left (x\right )\right ) \log \left (x\right ) - \frac {1}{30} \, {\left (22 \, x^{3} - 45 \, x^{2} - 30 \, {\left (x^{2} - 3 \, x\right )} e^{x} - 54 \, x\right )} \log \left (x\right ) - \frac {1}{30} \, {\left (2 \, x^{3} - 9 \, x^{2}\right )} \log \left (x\right ) \] Input:
integrate(1/5*((3*x^2-6*x)*log(x)+x^2-3*x)*log(x*log(x))+1/5*((5*x^2-5*x-1 5)*exp(x)-11*x^2+15*x+9)*log(x)+1/5*(5*x-15)*exp(x)-3/5*x^2+6/5*x+9/5,x, a lgorithm="maxima")
Output:
1/5*(x^3 - 3*x^2)*log(x*log(x))*log(x) - 1/30*(22*x^3 - 45*x^2 - 30*(x^2 - 3*x)*e^x - 54*x)*log(x) - 1/30*(2*x^3 - 9*x^2)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.82 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\frac {1}{5} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x\right )^{2} + \frac {1}{5} \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - {\left (x - 1\right )} e^{x} + {\left (x - 4\right )} e^{x} - \frac {1}{30} \, {\left (22 \, x^{3} - 45 \, x^{2} - 30 \, {\left (x^{2} - 3 \, x\right )} e^{x} - 54 \, x\right )} \log \left (x\right ) - \frac {1}{30} \, {\left (2 \, x^{3} - 9 \, x^{2}\right )} \log \left (x\right ) + 3 \, e^{x} \] Input:
integrate(1/5*((3*x^2-6*x)*log(x)+x^2-3*x)*log(x*log(x))+1/5*((5*x^2-5*x-1 5)*exp(x)-11*x^2+15*x+9)*log(x)+1/5*(5*x-15)*exp(x)-3/5*x^2+6/5*x+9/5,x, a lgorithm="giac")
Output:
1/5*(x^3 - 3*x^2)*log(x)^2 + 1/5*(x^3 - 3*x^2)*log(x)*log(log(x)) - (x - 1 )*e^x + (x - 4)*e^x - 1/30*(22*x^3 - 45*x^2 - 30*(x^2 - 3*x)*e^x - 54*x)*l og(x) - 1/30*(2*x^3 - 9*x^2)*log(x) + 3*e^x
Time = 3.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=-\frac {x\,\ln \left (x\right )\,\left (x-3\right )\,\left (4\,x-5\,{\mathrm {e}}^x-x\,\ln \left (x\,\ln \left (x\right )\right )+3\right )}{5} \] Input:
int((6*x)/5 + (log(x)*(15*x - exp(x)*(5*x - 5*x^2 + 15) - 11*x^2 + 9))/5 + (exp(x)*(5*x - 15))/5 - (log(x*log(x))*(3*x + log(x)*(6*x - 3*x^2) - x^2) )/5 - (3*x^2)/5 + 9/5,x)
Output:
-(x*log(x)*(x - 3)*(4*x - 5*exp(x) - x*log(x*log(x)) + 3))/5
Time = 0.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {1}{5} \left (9+6 x-3 x^2+e^x (-15+5 x)+\left (9+15 x-11 x^2+e^x \left (-15-5 x+5 x^2\right )\right ) \log (x)+\left (-3 x+x^2+\left (-6 x+3 x^2\right ) \log (x)\right ) \log (x \log (x))\right ) \, dx=\frac {\mathrm {log}\left (x \right ) x \left (5 e^{x} x -15 e^{x}+\mathrm {log}\left (\mathrm {log}\left (x \right ) x \right ) x^{2}-3 \,\mathrm {log}\left (\mathrm {log}\left (x \right ) x \right ) x -4 x^{2}+9 x +9\right )}{5} \] Input:
int(1/5*((3*x^2-6*x)*log(x)+x^2-3*x)*log(x*log(x))+1/5*((5*x^2-5*x-15)*exp (x)-11*x^2+15*x+9)*log(x)+1/5*(5*x-15)*exp(x)-3/5*x^2+6/5*x+9/5,x)
Output:
(log(x)*x*(5*e**x*x - 15*e**x + log(log(x)*x)*x**2 - 3*log(log(x)*x)*x - 4 *x**2 + 9*x + 9))/5