Integrand size = 229, antiderivative size = 31 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {4}{\left (-5-\frac {e^{2 x}}{5}+\frac {1}{x}-x\right ) \left (-\frac {x}{2}+\log (x)\right )} \end {dmath*}
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=-\frac {40 x}{\left (-5+25 x+e^{2 x} x+5 x^2\right ) (-x+2 \log (x))} \end {dmath*}
Integrate[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80 *x^3) + (400 + 400*x^2 + 160*E^(2*x)*x^2)*Log[x])/(25*x^2 - 250*x^3 + 575* x^4 + E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^5) + (-100*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^ (2*x)*(40*x^2 - 200*x^3 - 40*x^4))*Log[x] + (100 - 1000*x + 2300*x^2 + 4*E ^(4*x)*x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*Log[ x]^2),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.
\(\displaystyle \int \frac {-400 x^3-600 x^2+\left (160 e^{2 x} x^2+400 x^2+400\right ) \log (x)+e^{2 x} \left (-80 x^3-40 x^2+80 x\right )+2000 x-400}{25 x^6+250 x^5+e^{4 x} x^4+575 x^4-250 x^3+25 x^2+e^{2 x} \left (10 x^5+50 x^4-10 x^3\right )+\left (100 x^4+1000 x^3+4 e^{4 x} x^2+2300 x^2+e^{2 x} \left (40 x^3+200 x^2-40 x\right )-1000 x+100\right ) \log ^2(x)+\left (-100 x^5-1000 x^4-4 e^{4 x} x^3-2300 x^3+1000 x^2+e^{2 x} \left (-40 x^4-200 x^3+40 x^2\right )-100 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {40 \left (-2 \left (e^{2 x}+5\right ) x^3-\left (e^{2 x}+15\right ) x^2+2 \left (\left (2 e^{2 x}+5\right ) x^2+5\right ) \log (x)+2 \left (e^{2 x}+25\right ) x-10\right )}{\left (-5 x^2-\left (e^{2 x}+25\right ) x+5\right )^2 (x-2 \log (x))^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 40 \int -\frac {2 \left (5+e^{2 x}\right ) x^3+\left (15+e^{2 x}\right ) x^2-2 \left (25+e^{2 x}\right ) x-2 \left (\left (5+2 e^{2 x}\right ) x^2+5\right ) \log (x)+10}{\left (-5 x^2-\left (25+e^{2 x}\right ) x+5\right )^2 (x-2 \log (x))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -40 \int \frac {2 \left (5+e^{2 x}\right ) x^3+\left (15+e^{2 x}\right ) x^2-2 \left (25+e^{2 x}\right ) x-2 \left (\left (5+2 e^{2 x}\right ) x^2+5\right ) \log (x)+10}{\left (-5 x^2-\left (25+e^{2 x}\right ) x+5\right )^2 (x-2 \log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -40 \int \left (\frac {2 x^2-4 \log (x) x+x-2}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}-\frac {5 \left (2 x^3+9 x^2-2 x-1\right )}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -40 \left (-2 \int \frac {1}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+\int \frac {x}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+2 \int \frac {x^2}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx+5 \int \frac {1}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx+10 \int \frac {x}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx-45 \int \frac {x^2}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx-4 \int \frac {x \log (x)}{\left (5 x^2+e^{2 x} x+25 x-5\right ) (x-2 \log (x))^2}dx-10 \int \frac {x^3}{\left (5 x^2+e^{2 x} x+25 x-5\right )^2 (x-2 \log (x))}dx\right )\) |
Int[(-400 + 2000*x - 600*x^2 - 400*x^3 + E^(2*x)*(80*x - 40*x^2 - 80*x^3) + (400 + 400*x^2 + 160*E^(2*x)*x^2)*Log[x])/(25*x^2 - 250*x^3 + 575*x^4 + E^(4*x)*x^4 + 250*x^5 + 25*x^6 + E^(2*x)*(-10*x^3 + 50*x^4 + 10*x^5) + (-1 00*x + 1000*x^2 - 2300*x^3 - 4*E^(4*x)*x^3 - 1000*x^4 - 100*x^5 + E^(2*x)* (40*x^2 - 200*x^3 - 40*x^4))*Log[x] + (100 - 1000*x + 2300*x^2 + 4*E^(4*x) *x^2 + 1000*x^3 + 100*x^4 + E^(2*x)*(-40*x + 200*x^2 + 40*x^3))*Log[x]^2), x]
3.8.65.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {40 x}{\left (5 x^{2}+x \,{\mathrm e}^{2 x}+25 x -5\right ) \left (x -2 \ln \left (x \right )\right )}\) | \(30\) |
parallelrisch | \(\frac {40 x}{5 x^{3}+{\mathrm e}^{2 x} x^{2}-10 x^{2} \ln \left (x \right )-2 \ln \left (x \right ) {\mathrm e}^{2 x} x +25 x^{2}-50 x \ln \left (x \right )-5 x +10 \ln \left (x \right )}\) | \(53\) |
int(((160*exp(2*x)*x^2+400*x^2+400)*ln(x)+(-80*x^3-40*x^2+80*x)*exp(2*x)-4 00*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x)*exp(2* x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*ln(x)^2+(-4*x^3*exp(2*x)^2+(-40*x ^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100*x)*ln(x )+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+575*x^4-25 0*x^3+25*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 25 \, x^{2} - 2 \, {\left (5 \, x^{2} + x e^{\left (2 \, x\right )} + 25 \, x - 5\right )} \log \left (x\right ) - 5 \, x} \end {dmath*}
integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp (2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) *exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 *x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 75*x^4-250*x^3+25*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 x}{5 x^{3} - 10 x^{2} \log {\left (x \right )} + 25 x^{2} - 50 x \log {\left (x \right )} - 5 x + \left (x^{2} - 2 x \log {\left (x \right )}\right ) e^{2 x} + 10 \log {\left (x \right )}} \end {dmath*}
integrate(((160*exp(2*x)*x**2+400*x**2+400)*ln(x)+(-80*x**3-40*x**2+80*x)* exp(2*x)-400*x**3-600*x**2+2000*x-400)/((4*x**2*exp(2*x)**2+(40*x**3+200*x **2-40*x)*exp(2*x)+100*x**4+1000*x**3+2300*x**2-1000*x+100)*ln(x)**2+(-4*x **3*exp(2*x)**2+(-40*x**4-200*x**3+40*x**2)*exp(2*x)-100*x**5-1000*x**4-23 00*x**3+1000*x**2-100*x)*ln(x)+x**4*exp(2*x)**2+(10*x**5+50*x**4-10*x**3)* exp(2*x)+25*x**6+250*x**5+575*x**4-250*x**3+25*x**2),x)
40*x/(5*x**3 - 10*x**2*log(x) + 25*x**2 - 50*x*log(x) - 5*x + (x**2 - 2*x* log(x))*exp(2*x) + 10*log(x))
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + 25 \, x^{2} + {\left (x^{2} - 2 \, x \log \left (x\right )\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{2} + 5 \, x - 1\right )} \log \left (x\right ) - 5 \, x} \end {dmath*}
integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp (2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) *exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 *x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 75*x^4-250*x^3+25*x^2),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=\frac {40 \, x}{5 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 10 \, x^{2} \log \left (x\right ) - 2 \, x e^{\left (2 \, x\right )} \log \left (x\right ) + 25 \, x^{2} - 50 \, x \log \left (x\right ) - 5 \, x + 10 \, \log \left (x\right )} \end {dmath*}
integrate(((160*exp(2*x)*x^2+400*x^2+400)*log(x)+(-80*x^3-40*x^2+80*x)*exp (2*x)-400*x^3-600*x^2+2000*x-400)/((4*x^2*exp(2*x)^2+(40*x^3+200*x^2-40*x) *exp(2*x)+100*x^4+1000*x^3+2300*x^2-1000*x+100)*log(x)^2+(-4*x^3*exp(2*x)^ 2+(-40*x^4-200*x^3+40*x^2)*exp(2*x)-100*x^5-1000*x^4-2300*x^3+1000*x^2-100 *x)*log(x)+x^4*exp(2*x)^2+(10*x^5+50*x^4-10*x^3)*exp(2*x)+25*x^6+250*x^5+5 75*x^4-250*x^3+25*x^2),x, algorithm=\
40*x/(5*x^3 + x^2*e^(2*x) - 10*x^2*log(x) - 2*x*e^(2*x)*log(x) + 25*x^2 - 50*x*log(x) - 5*x + 10*log(x))
Timed out. \begin {dmath*} \int \frac {-400+2000 x-600 x^2-400 x^3+e^{2 x} \left (80 x-40 x^2-80 x^3\right )+\left (400+400 x^2+160 e^{2 x} x^2\right ) \log (x)}{25 x^2-250 x^3+575 x^4+e^{4 x} x^4+250 x^5+25 x^6+e^{2 x} \left (-10 x^3+50 x^4+10 x^5\right )+\left (-100 x+1000 x^2-2300 x^3-4 e^{4 x} x^3-1000 x^4-100 x^5+e^{2 x} \left (40 x^2-200 x^3-40 x^4\right )\right ) \log (x)+\left (100-1000 x+2300 x^2+4 e^{4 x} x^2+1000 x^3+100 x^4+e^{2 x} \left (-40 x+200 x^2+40 x^3\right )\right ) \log ^2(x)} \, dx=-\int \frac {{\mathrm {e}}^{2\,x}\,\left (80\,x^3+40\,x^2-80\,x\right )-2000\,x-\ln \left (x\right )\,\left (160\,x^2\,{\mathrm {e}}^{2\,x}+400\,x^2+400\right )+600\,x^2+400\,x^3+400}{{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (40\,x^3+200\,x^2-40\,x\right )-1000\,x+4\,x^2\,{\mathrm {e}}^{4\,x}+2300\,x^2+1000\,x^3+100\,x^4+100\right )-\ln \left (x\right )\,\left (100\,x+4\,x^3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (40\,x^4+200\,x^3-40\,x^2\right )-1000\,x^2+2300\,x^3+1000\,x^4+100\,x^5\right )+x^4\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{2\,x}\,\left (10\,x^5+50\,x^4-10\,x^3\right )+25\,x^2-250\,x^3+575\,x^4+250\,x^5+25\,x^6} \,d x \end {dmath*}
int(-(exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x ) + 400*x^2 + 400) + 600*x^2 + 400*x^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*x^ 4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 4 0*x^4) - 1000*x^2 + 2300*x^3 + 1000*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2* x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 + 25* x^6),x)
-int((exp(2*x)*(40*x^2 - 80*x + 80*x^3) - 2000*x - log(x)*(160*x^2*exp(2*x ) + 400*x^2 + 400) + 600*x^2 + 400*x^3 + 400)/(log(x)^2*(exp(2*x)*(200*x^2 - 40*x + 40*x^3) - 1000*x + 4*x^2*exp(4*x) + 2300*x^2 + 1000*x^3 + 100*x^ 4 + 100) - log(x)*(100*x + 4*x^3*exp(4*x) + exp(2*x)*(200*x^3 - 40*x^2 + 4 0*x^4) - 1000*x^2 + 2300*x^3 + 1000*x^4 + 100*x^5) + x^4*exp(4*x) + exp(2* x)*(50*x^4 - 10*x^3 + 10*x^5) + 25*x^2 - 250*x^3 + 575*x^4 + 250*x^5 + 25* x^6), x)