Integrand size = 125, antiderivative size = 30 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {1}{16} x^2 \left (x+5 \left (2 \left (x-x^2-x^x\right )+\log (x)\right )^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(30)=60\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {1}{16} x^2 \left (x+20 x^2-40 x^3+20 x^4+20 x^{2 x}-40 x^{1+x}+40 x^{2+x}-20 \left (-x+x^2+x^x\right ) \log (x)+5 \log ^2(x)\right ) \]
Integrate[(23*x^2 + 60*x^3 - 200*x^4 + 120*x^5 + (10*x + 60*x^2 - 80*x^3)* Log[x] + 10*x*Log[x]^2 + x^(2*x)*(40*x + 40*x^2 + 40*x^2*Log[x]) + x^x*(-2 0*x - 120*x^2 + 120*x^3 + 40*x^4 + (-40*x - 20*x^2 - 40*x^3 + 40*x^4)*Log[ x] - 20*x^2*Log[x]^2))/16,x]
(x^2*(x + 20*x^2 - 40*x^3 + 20*x^4 + 20*x^(2*x) - 40*x^(1 + x) + 40*x^(2 + x) - 20*(-x + x^2 + x^x)*Log[x] + 5*Log[x]^2))/16
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.
\(\displaystyle \int \frac {1}{16} \left (120 x^5-200 x^4+60 x^3+23 x^2+x^{2 x} \left (40 x^2+40 x^2 \log (x)+40 x\right )+\left (-80 x^3+60 x^2+10 x\right ) \log (x)+x^x \left (40 x^4+120 x^3-120 x^2-20 x^2 \log ^2(x)+\left (40 x^4-40 x^3-20 x^2-40 x\right ) \log (x)-20 x\right )+10 x \log ^2(x)\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{16} \int \left (-20 \left (-2 x^4-6 x^3+\log ^2(x) x^2+6 x^2+x+\left (-2 x^4+2 x^3+x^2+2 x\right ) \log (x)\right ) x^x+40 \left (\log (x) x^2+x^2+x\right ) x^{2 x}+120 x^5-200 x^4+60 x^3+23 x^2+10 \log ^2(x) x+10 \left (-8 x^3+6 x^2+x\right ) \log (x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{16} \left (-20 \int x^{x+1}dx-120 \int x^{x+2}dx+120 \int x^{x+3}dx+40 \int x^{x+4}dx+40 \int x^{2 x+1}dx+40 \int x^{2 x+2}dx+40 \int \frac {\int x^{x+1}dx}{x}dx+20 \int \frac {\int x^{x+2}dx}{x}dx+40 \int \frac {\int x^{x+3}dx}{x}dx-40 \int \frac {\int x^{x+4}dx}{x}dx-40 \int \frac {\int x^{2 x+2}dx}{x}dx-20 \int x^{x+2} \log ^2(x)dx-40 \log (x) \int x^{x+1}dx-20 \log (x) \int x^{x+2}dx-40 \log (x) \int x^{x+3}dx+40 \log (x) \int x^{x+4}dx+40 \log (x) \int x^{2 x+2}dx+20 x^6-40 x^5+20 x^4-20 x^4 \log (x)+x^3+20 x^3 \log (x)+5 x^2 \log ^2(x)\right )\) |
Int[(23*x^2 + 60*x^3 - 200*x^4 + 120*x^5 + (10*x + 60*x^2 - 80*x^3)*Log[x] + 10*x*Log[x]^2 + x^(2*x)*(40*x + 40*x^2 + 40*x^2*Log[x]) + x^x*(-20*x - 120*x^2 + 120*x^3 + 40*x^4 + (-40*x - 20*x^2 - 40*x^3 + 40*x^4)*Log[x] - 2 0*x^2*Log[x]^2))/16,x]
3.2.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(29)=58\).
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.97
method | result | size |
parallelrisch | \(\frac {5 \,{\mathrm e}^{2 x \ln \left (x \right )} x^{2}}{4}+\frac {5 \,{\mathrm e}^{x \ln \left (x \right )} x^{4}}{2}-\frac {5 \,{\mathrm e}^{x \ln \left (x \right )} \ln \left (x \right ) x^{2}}{4}-\frac {5 \,{\mathrm e}^{x \ln \left (x \right )} x^{3}}{2}+\frac {5 x^{2} \ln \left (x \right )^{2}}{16}-\frac {5 x^{4} \ln \left (x \right )}{4}+\frac {5 x^{3} \ln \left (x \right )}{4}+\frac {5 x^{4}}{4}+\frac {x^{3}}{16}+\frac {5 x^{6}}{4}-\frac {5 x^{5}}{2}\) | \(89\) |
risch | \(\frac {5 x^{2 x} x^{2}}{4}+\frac {\left (40 x^{4}-40 x^{3}-20 x^{2} \ln \left (x \right )\right ) x^{x}}{16}+\frac {5 x^{2} \ln \left (x \right )^{2}}{16}-\frac {5 x^{2} \ln \left (x \right )}{16}+\frac {\left (-20 x^{4}+20 x^{3}+5 x^{2}\right ) \ln \left (x \right )}{16}+\frac {5 x^{4}}{4}+\frac {x^{3}}{16}+\frac {5 x^{6}}{4}-\frac {5 x^{5}}{2}\) | \(91\) |
int(1/16*(40*x^2*ln(x)+40*x^2+40*x)*exp(x*ln(x))^2+1/16*(-20*x^2*ln(x)^2+( 40*x^4-40*x^3-20*x^2-40*x)*ln(x)+40*x^4+120*x^3-120*x^2-20*x)*exp(x*ln(x)) +5/8*x*ln(x)^2+1/16*(-80*x^3+60*x^2+10*x)*ln(x)+15/2*x^5-25/2*x^4+15/4*x^3 +23/16*x^2,x,method=_RETURNVERBOSE)
5/4*exp(x*ln(x))^2*x^2+5/2*exp(x*ln(x))*x^4-5/4*exp(x*ln(x))*ln(x)*x^2-5/2 *exp(x*ln(x))*x^3+5/16*x^2*ln(x)^2-5/4*x^4*ln(x)+5/4*x^3*ln(x)+5/4*x^4+1/1 6*x^3+5/4*x^6-5/2*x^5
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {5}{4} \, x^{6} - \frac {5}{2} \, x^{5} + \frac {5}{4} \, x^{4} + \frac {5}{16} \, x^{2} \log \left (x\right )^{2} + \frac {5}{4} \, x^{2} x^{2 \, x} + \frac {1}{16} \, x^{3} + \frac {5}{4} \, {\left (2 \, x^{4} - 2 \, x^{3} - x^{2} \log \left (x\right )\right )} x^{x} - \frac {5}{4} \, {\left (x^{4} - x^{3}\right )} \log \left (x\right ) \]
integrate(1/16*(40*x^2*log(x)+40*x^2+40*x)*exp(x*log(x))^2+1/16*(-20*x^2*l og(x)^2+(40*x^4-40*x^3-20*x^2-40*x)*log(x)+40*x^4+120*x^3-120*x^2-20*x)*ex p(x*log(x))+5/8*x*log(x)^2+1/16*(-80*x^3+60*x^2+10*x)*log(x)+15/2*x^5-25/2 *x^4+15/4*x^3+23/16*x^2,x, algorithm=\
5/4*x^6 - 5/2*x^5 + 5/4*x^4 + 5/16*x^2*log(x)^2 + 5/4*x^2*x^(2*x) + 1/16*x ^3 + 5/4*(2*x^4 - 2*x^3 - x^2*log(x))*x^x - 5/4*(x^4 - x^3)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (29) = 58\).
Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.17 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {5 x^{6}}{4} - \frac {5 x^{5}}{2} + \frac {5 x^{4}}{4} + \frac {x^{3}}{16} + \frac {5 x^{2} e^{2 x \log {\left (x \right )}}}{4} + \frac {5 x^{2} \log {\left (x \right )}^{2}}{16} + \left (- \frac {5 x^{4}}{4} + \frac {5 x^{3}}{4}\right ) \log {\left (x \right )} + \frac {\left (40 x^{4} - 40 x^{3} - 20 x^{2} \log {\left (x \right )}\right ) e^{x \log {\left (x \right )}}}{16} \]
integrate(1/16*(40*x**2*ln(x)+40*x**2+40*x)*exp(x*ln(x))**2+1/16*(-20*x**2 *ln(x)**2+(40*x**4-40*x**3-20*x**2-40*x)*ln(x)+40*x**4+120*x**3-120*x**2-2 0*x)*exp(x*ln(x))+5/8*x*ln(x)**2+1/16*(-80*x**3+60*x**2+10*x)*ln(x)+15/2*x **5-25/2*x**4+15/4*x**3+23/16*x**2,x)
5*x**6/4 - 5*x**5/2 + 5*x**4/4 + x**3/16 + 5*x**2*exp(2*x*log(x))/4 + 5*x* *2*log(x)**2/16 + (-5*x**4/4 + 5*x**3/4)*log(x) + (40*x**4 - 40*x**3 - 20* x**2*log(x))*exp(x*log(x))/16
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (29) = 58\).
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {5}{4} \, x^{6} - \frac {5}{2} \, x^{5} + \frac {5}{4} \, x^{4} + \frac {5}{4} \, x^{2} x^{2 \, x} + \frac {5}{32} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + \frac {1}{16} \, x^{3} + \frac {5}{4} \, {\left (2 \, x^{4} - 2 \, x^{3} - x^{2} \log \left (x\right )\right )} x^{x} - \frac {5}{32} \, x^{2} - \frac {5}{16} \, {\left (4 \, x^{4} - 4 \, x^{3} - x^{2}\right )} \log \left (x\right ) \]
integrate(1/16*(40*x^2*log(x)+40*x^2+40*x)*exp(x*log(x))^2+1/16*(-20*x^2*l og(x)^2+(40*x^4-40*x^3-20*x^2-40*x)*log(x)+40*x^4+120*x^3-120*x^2-20*x)*ex p(x*log(x))+5/8*x*log(x)^2+1/16*(-80*x^3+60*x^2+10*x)*log(x)+15/2*x^5-25/2 *x^4+15/4*x^3+23/16*x^2,x, algorithm=\
5/4*x^6 - 5/2*x^5 + 5/4*x^4 + 5/4*x^2*x^(2*x) + 5/32*(2*log(x)^2 - 2*log(x ) + 1)*x^2 + 1/16*x^3 + 5/4*(2*x^4 - 2*x^3 - x^2*log(x))*x^x - 5/32*x^2 - 5/16*(4*x^4 - 4*x^3 - x^2)*log(x)
Timed out. \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\text {Timed out} \]
integrate(1/16*(40*x^2*log(x)+40*x^2+40*x)*exp(x*log(x))^2+1/16*(-20*x^2*l og(x)^2+(40*x^4-40*x^3-20*x^2-40*x)*log(x)+40*x^4+120*x^3-120*x^2-20*x)*ex p(x*log(x))+5/8*x*log(x)^2+1/16*(-80*x^3+60*x^2+10*x)*log(x)+15/2*x^5-25/2 *x^4+15/4*x^3+23/16*x^2,x, algorithm=\
Time = 11.66 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {1}{16} \left (23 x^2+60 x^3-200 x^4+120 x^5+\left (10 x+60 x^2-80 x^3\right ) \log (x)+10 x \log ^2(x)+x^{2 x} \left (40 x+40 x^2+40 x^2 \log (x)\right )+x^x \left (-20 x-120 x^2+120 x^3+40 x^4+\left (-40 x-20 x^2-40 x^3+40 x^4\right ) \log (x)-20 x^2 \log ^2(x)\right )\right ) \, dx=\frac {x^2\,\left (x-20\,x^2\,\ln \left (x\right )-40\,x\,x^x+5\,{\ln \left (x\right )}^2+40\,x^x\,x^2-20\,x^x\,\ln \left (x\right )+20\,x^{2\,x}+20\,x\,\ln \left (x\right )+20\,x^2-40\,x^3+20\,x^4\right )}{16} \]
int((exp(2*x*log(x))*(40*x + 40*x^2*log(x) + 40*x^2))/16 + (5*x*log(x)^2)/ 8 + (23*x^2)/16 + (15*x^3)/4 - (25*x^4)/2 + (15*x^5)/2 + (log(x)*(10*x + 6 0*x^2 - 80*x^3))/16 - (exp(x*log(x))*(20*x + log(x)*(40*x + 20*x^2 + 40*x^ 3 - 40*x^4) + 20*x^2*log(x)^2 + 120*x^2 - 120*x^3 - 40*x^4))/16,x)