Integrand size = 198, antiderivative size = 24 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\frac {2 \log (x) \left (\left (-1-e^{x/5}+x\right )^4+\log (x)\right )}{x} \] Output:
2*(ln(x)+(x-exp(1/5*x)-1)^4)/x*ln(x)
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\frac {2 \log (x) \left (\left (1+e^{x/5}-x\right )^4+\log (x)\right )}{x} \] Input:
Integrate[(10 + 10*E^((4*x)/5) + E^((3*x)/5)*(40 - 40*x) - 40*x + 60*x^2 - 40*x^3 + 10*x^4 + E^((2*x)/5)*(60 - 120*x + 60*x^2) + E^(x/5)*(40 - 120*x + 120*x^2 - 40*x^3) + (10 + 60*x^2 - 80*x^3 + 30*x^4 + E^((4*x)/5)*(-10 + 8*x) + E^((3*x)/5)*(-40 + 24*x - 24*x^2) + E^((2*x)/5)*(-60 + 24*x + 12*x ^2 + 24*x^3) + E^(x/5)*(-40 + 8*x + 96*x^2 - 56*x^3 - 8*x^4))*Log[x] - 10* Log[x]^2)/(5*x^2),x]
Output:
(2*Log[x]*((1 + E^(x/5) - x)^4 + Log[x]))/x
Leaf count is larger than twice the leaf count of optimal. \(170\) vs. \(2(24)=48\).
Time = 1.70 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, 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.
| \(\displaystyle \int \frac {10 x^4-40 x^3+60 x^2+e^{2 x/5} \left (60 x^2-120 x+60\right )+e^{x/5} \left (-40 x^3+120 x^2-120 x+40\right )+\left (30 x^4-80 x^3+60 x^2+e^{3 x/5} \left (-24 x^2+24 x-40\right )+e^{2 x/5} \left (24 x^3+12 x^2+24 x-60\right )+e^{x/5} \left (-8 x^4-56 x^3+96 x^2+8 x-40\right )+e^{4 x/5} (8 x-10)+10\right ) \log (x)-40 x+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-10 \log ^2(x)+10}{5 x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {2 \left (5 x^4-20 x^3+30 x^2-20 x+5 e^{4 x/5}-5 \log ^2(x)+20 e^{3 x/5} (1-x)+30 e^{2 x/5} \left (x^2-2 x+1\right )+20 e^{x/5} \left (-x^3+3 x^2-3 x+1\right )+\left (15 x^4-40 x^3+30 x^2-e^{4 x/5} (5-4 x)-4 e^{3 x/5} \left (3 x^2-3 x+5\right )-6 e^{2 x/5} \left (-2 x^3-x^2-2 x+5\right )-4 e^{x/5} \left (x^4+7 x^3-12 x^2-x+5\right )+5\right ) \log (x)+5\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} \int \frac {5 x^4-20 x^3+30 x^2-20 x+5 e^{4 x/5}-5 \log ^2(x)+20 e^{3 x/5} (1-x)+30 e^{2 x/5} \left (x^2-2 x+1\right )+20 e^{x/5} \left (-x^3+3 x^2-3 x+1\right )+\left (15 x^4-40 x^3+30 x^2-e^{4 x/5} (5-4 x)-4 e^{3 x/5} \left (3 x^2-3 x+5\right )-6 e^{2 x/5} \left (-2 x^3-x^2-2 x+5\right )-4 e^{x/5} \left (x^4+7 x^3-12 x^2-x+5\right )+5\right ) \log (x)+5}{x^2}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {2}{5} \int \left (-\frac {4 e^{x/5} \left (\log (x) x^2+9 \log (x) x+5 x+5 \log (x)-5\right ) (x-1)^2}{x^2}+\frac {6 e^{2 x/5} \left (2 \log (x) x^2+3 \log (x) x+5 x+5 \log (x)-5\right ) (x-1)}{x^2}+\frac {e^{4 x/5} (4 x \log (x)-5 \log (x)+5)}{x^2}-\frac {4 e^{3 x/5} \left (3 \log (x) x^2-3 \log (x) x+5 x+5 \log (x)-5\right )}{x^2}+\frac {5 \left (3 \log (x) x^4+x^4-8 \log (x) x^3-4 x^3+6 \log (x) x^2+6 x^2-4 x-\log ^2(x)+\log (x)+1\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} \left (5 x^3 \log (x)-20 e^{x/5} x^2 \log (x)-20 x^2 \log (x)+\frac {20 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}+\frac {5 \log ^2(x)}{x}+60 e^{x/5} x \log (x)+30 e^{2 x/5} x \log (x)+30 x \log (x)-60 e^{x/5} \log (x)-60 e^{2 x/5} \log (x)-20 \log (x)+\frac {20 e^{x/5} \log (x)}{x}+\frac {30 e^{2 x/5} \log (x)}{x}+\frac {5 e^{4 x/5} \log (x)}{x}+\frac {5 \log (x)}{x}\right )\) |
Input:
Int[(10 + 10*E^((4*x)/5) + E^((3*x)/5)*(40 - 40*x) - 40*x + 60*x^2 - 40*x^ 3 + 10*x^4 + E^((2*x)/5)*(60 - 120*x + 60*x^2) + E^(x/5)*(40 - 120*x + 120 *x^2 - 40*x^3) + (10 + 60*x^2 - 80*x^3 + 30*x^4 + E^((4*x)/5)*(-10 + 8*x) + E^((3*x)/5)*(-40 + 24*x - 24*x^2) + E^((2*x)/5)*(-60 + 24*x + 12*x^2 + 2 4*x^3) + E^(x/5)*(-40 + 8*x + 96*x^2 - 56*x^3 - 8*x^4))*Log[x] - 10*Log[x] ^2)/(5*x^2),x]
Output:
(2*(-20*Log[x] - 60*E^(x/5)*Log[x] - 60*E^((2*x)/5)*Log[x] + (5*Log[x])/x + (20*E^(x/5)*Log[x])/x + (30*E^((2*x)/5)*Log[x])/x + (5*E^((4*x)/5)*Log[x ])/x + 30*x*Log[x] + 60*E^(x/5)*x*Log[x] + 30*E^((2*x)/5)*x*Log[x] - 20*x^ 2*Log[x] - 20*E^(x/5)*x^2*Log[x] + 5*x^3*Log[x] + (5*Log[x]^2)/x + (20*E^( (3*x)/5)*(x*Log[x] - x^2*Log[x]))/x^2))/5
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. \(153\) vs. \(2(21)=42\).
Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 6.42
\[\frac {2 \ln \left (x \right ) {\mathrm e}^{\frac {4 x}{5}}}{x}+\frac {40 \ln \left (x \right ) {\mathrm e}^{\frac {3 x}{5}}-40 \ln \left (x \right ) {\mathrm e}^{\frac {3 x}{5}} x}{5 x}+\frac {60 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}}-120 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}} x +60 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}} x^{2}}{5 x}+\frac {40 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}}-120 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}} x +120 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}} x^{2}-40 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}} x^{3}}{5 x}-8 \ln \left (x \right )+\frac {2 \ln \left (x \right )^{2}}{x}+\frac {2 \ln \left (x \right )}{x}+2 x^{3} \ln \left (x \right )-8 x^{2} \ln \left (x \right )+12 x \ln \left (x \right )\]
Input:
int(1/5*(-10*ln(x)^2+((8*x-10)*exp(1/5*x)^4+(-24*x^2+24*x-40)*exp(1/5*x)^3 +(24*x^3+12*x^2+24*x-60)*exp(1/5*x)^2+(-8*x^4-56*x^3+96*x^2+8*x-40)*exp(1/ 5*x)+30*x^4-80*x^3+60*x^2+10)*ln(x)+10*exp(1/5*x)^4+(-40*x+40)*exp(1/5*x)^ 3+(60*x^2-120*x+60)*exp(1/5*x)^2+(-40*x^3+120*x^2-120*x+40)*exp(1/5*x)+10* x^4-40*x^3+60*x^2-40*x+10)/x^2,x)
Output:
2/x*ln(x)*exp(4/5*x)+1/5*(40*ln(x)*exp(3/5*x)-40*ln(x)*exp(3/5*x)*x)/x+1/5 *(60*ln(x)*exp(2/5*x)-120*ln(x)*exp(2/5*x)*x+60*ln(x)*exp(2/5*x)*x^2)/x+1/ 5*(40*ln(x)*exp(1/5*x)-120*ln(x)*exp(1/5*x)*x+120*ln(x)*exp(1/5*x)*x^2-40* ln(x)*exp(1/5*x)*x^3)/x-8*ln(x)+2*ln(x)^2/x+2*ln(x)/x+2*x^3*ln(x)-8*x^2*ln (x)+12*x*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.21 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\frac {2 \, {\left ({\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, {\left (x - 1\right )} e^{\left (\frac {3}{5} \, x\right )} + 6 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (\frac {2}{5} \, x\right )} - 4 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (\frac {1}{5} \, x\right )} - 4 \, x + e^{\left (\frac {4}{5} \, x\right )} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{x} \] Input:
integrate(1/5*(-10*log(x)^2+((8*x-10)*exp(1/5*x)^4+(-24*x^2+24*x-40)*exp(1 /5*x)^3+(24*x^3+12*x^2+24*x-60)*exp(1/5*x)^2+(-8*x^4-56*x^3+96*x^2+8*x-40) *exp(1/5*x)+30*x^4-80*x^3+60*x^2+10)*log(x)+10*exp(1/5*x)^4+(-40*x+40)*exp (1/5*x)^3+(60*x^2-120*x+60)*exp(1/5*x)^2+(-40*x^3+120*x^2-120*x+40)*exp(1/ 5*x)+10*x^4-40*x^3+60*x^2-40*x+10)/x^2,x, algorithm="fricas")
Output:
2*((x^4 - 4*x^3 + 6*x^2 - 4*(x - 1)*e^(3/5*x) + 6*(x^2 - 2*x + 1)*e^(2/5*x ) - 4*(x^3 - 3*x^2 + 3*x - 1)*e^(1/5*x) - 4*x + e^(4/5*x) + 1)*log(x) + lo g(x)^2)/x
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 6.17 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=- 8 \log {\left (x \right )} + \frac {\left (2 x^{4} - 8 x^{3} + 12 x^{2} + 2\right ) \log {\left (x \right )}}{x} + \frac {2 \log {\left (x \right )}^{2}}{x} + \frac {2 x^{3} e^{\frac {4 x}{5}} \log {\left (x \right )} + \left (- 8 x^{4} \log {\left (x \right )} + 8 x^{3} \log {\left (x \right )}\right ) e^{\frac {3 x}{5}} + \left (12 x^{5} \log {\left (x \right )} - 24 x^{4} \log {\left (x \right )} + 12 x^{3} \log {\left (x \right )}\right ) e^{\frac {2 x}{5}} + \left (- 8 x^{6} \log {\left (x \right )} + 24 x^{5} \log {\left (x \right )} - 24 x^{4} \log {\left (x \right )} + 8 x^{3} \log {\left (x \right )}\right ) e^{\frac {x}{5}}}{x^{4}} \] Input:
integrate(1/5*(-10*ln(x)**2+((8*x-10)*exp(1/5*x)**4+(-24*x**2+24*x-40)*exp (1/5*x)**3+(24*x**3+12*x**2+24*x-60)*exp(1/5*x)**2+(-8*x**4-56*x**3+96*x** 2+8*x-40)*exp(1/5*x)+30*x**4-80*x**3+60*x**2+10)*ln(x)+10*exp(1/5*x)**4+(- 40*x+40)*exp(1/5*x)**3+(60*x**2-120*x+60)*exp(1/5*x)**2+(-40*x**3+120*x**2 -120*x+40)*exp(1/5*x)+10*x**4-40*x**3+60*x**2-40*x+10)/x**2,x)
Output:
-8*log(x) + (2*x**4 - 8*x**3 + 12*x**2 + 2)*log(x)/x + 2*log(x)**2/x + (2* x**3*exp(4*x/5)*log(x) + (-8*x**4*log(x) + 8*x**3*log(x))*exp(3*x/5) + (12 *x**5*log(x) - 24*x**4*log(x) + 12*x**3*log(x))*exp(2*x/5) + (-8*x**6*log( x) + 24*x**5*log(x) - 24*x**4*log(x) + 8*x**3*log(x))*exp(x/5))/x**4
\[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\int { \frac {2 \, {\left (5 \, x^{4} - 20 \, x^{3} + 30 \, x^{2} - 20 \, {\left (x - 1\right )} e^{\left (\frac {3}{5} \, x\right )} + 30 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (\frac {2}{5} \, x\right )} - 20 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (\frac {1}{5} \, x\right )} + {\left (15 \, x^{4} - 40 \, x^{3} + 30 \, x^{2} + {\left (4 \, x - 5\right )} e^{\left (\frac {4}{5} \, x\right )} - 4 \, {\left (3 \, x^{2} - 3 \, x + 5\right )} e^{\left (\frac {3}{5} \, x\right )} + 6 \, {\left (2 \, x^{3} + x^{2} + 2 \, x - 5\right )} e^{\left (\frac {2}{5} \, x\right )} - 4 \, {\left (x^{4} + 7 \, x^{3} - 12 \, x^{2} - x + 5\right )} e^{\left (\frac {1}{5} \, x\right )} + 5\right )} \log \left (x\right ) - 5 \, \log \left (x\right )^{2} - 20 \, x + 5 \, e^{\left (\frac {4}{5} \, x\right )} + 5\right )}}{5 \, x^{2}} \,d x } \] Input:
integrate(1/5*(-10*log(x)^2+((8*x-10)*exp(1/5*x)^4+(-24*x^2+24*x-40)*exp(1 /5*x)^3+(24*x^3+12*x^2+24*x-60)*exp(1/5*x)^2+(-8*x^4-56*x^3+96*x^2+8*x-40) *exp(1/5*x)+30*x^4-80*x^3+60*x^2+10)*log(x)+10*exp(1/5*x)^4+(-40*x+40)*exp (1/5*x)^3+(60*x^2-120*x+60)*exp(1/5*x)^2+(-40*x^3+120*x^2-120*x+40)*exp(1/ 5*x)+10*x^4-40*x^3+60*x^2-40*x+10)/x^2,x, algorithm="maxima")
Output:
2*x^3*log(x) - 8*x^2*log(x) - 40*(x - 5)*e^(1/5*x) + 12*x*log(x) + 96*e^(1 /5*x)*log(x) - 2*(4*(x - 1)*e^(3/5*x)*log(x) - 6*(x^2 - 2*x + 1)*e^(2/5*x) *log(x) + 4*(x^3 - 3*x^2 + 15*x - 1)*e^(1/5*x)*log(x) - e^(4/5*x)*log(x) - log(x)^2 - 2*log(x) - 2)/x - 2*log(x)/x - 4/x - 8*Ei(3/5*x) - 24*Ei(2/5*x ) - 120*Ei(1/5*x) + 30*e^(2/5*x) + 120*e^(1/5*x) + 8/5*gamma(-1, -1/5*x) + 24/5*gamma(-1, -2/5*x) + 24/5*gamma(-1, -3/5*x) + 8/5*gamma(-1, -4/5*x) + 2/5*integrate(20*(x - 1)*e^(3/5*x)/x^2, x) - 2/5*integrate(30*(x^2 - 2*x + 1)*e^(2/5*x)/x^2, x) + 2/5*integrate(20*(x^3 - 3*x^2 + 15*x - 1)*e^(1/5* x)/x^2, x) - 2*integrate(e^(4/5*x)/x^2, x) - 8*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (21) = 42\).
Time = 0.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 11.88 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\frac {2 \, {\left (x^{4} \log \left (5\right ) - 4 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + x^{4} \log \left (\frac {1}{5} \, x\right ) - 4 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 4 \, x^{3} \log \left (5\right ) + 6 \, x^{2} e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) + 12 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) - 4 \, x^{3} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{2} e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 12 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{2} \log \left (5\right ) - 4 \, x e^{\left (\frac {3}{5} \, x\right )} \log \left (5\right ) - 12 \, x e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) - 12 \, x e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + 6 \, x^{2} \log \left (\frac {1}{5} \, x\right ) - 4 \, x e^{\left (\frac {3}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 12 \, x e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 12 \, x e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + e^{\left (\frac {4}{5} \, x\right )} \log \left (5\right ) + 4 \, e^{\left (\frac {3}{5} \, x\right )} \log \left (5\right ) + 6 \, e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) + 4 \, e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 4 \, x \log \left (\frac {1}{5} \, x\right ) + e^{\left (\frac {4}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 4 \, e^{\left (\frac {3}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 6 \, e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 4 \, e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 2 \, \log \left (5\right ) \log \left (\frac {1}{5} \, x\right ) + \log \left (\frac {1}{5} \, x\right )^{2} + \log \left (5\right ) + \log \left (\frac {1}{5} \, x\right )\right )}}{x} \] Input:
integrate(1/5*(-10*log(x)^2+((8*x-10)*exp(1/5*x)^4+(-24*x^2+24*x-40)*exp(1 /5*x)^3+(24*x^3+12*x^2+24*x-60)*exp(1/5*x)^2+(-8*x^4-56*x^3+96*x^2+8*x-40) *exp(1/5*x)+30*x^4-80*x^3+60*x^2+10)*log(x)+10*exp(1/5*x)^4+(-40*x+40)*exp (1/5*x)^3+(60*x^2-120*x+60)*exp(1/5*x)^2+(-40*x^3+120*x^2-120*x+40)*exp(1/ 5*x)+10*x^4-40*x^3+60*x^2-40*x+10)/x^2,x, algorithm="giac")
Output:
2*(x^4*log(5) - 4*x^3*e^(1/5*x)*log(5) + x^4*log(1/5*x) - 4*x^3*e^(1/5*x)* log(1/5*x) - 4*x^3*log(5) + 6*x^2*e^(2/5*x)*log(5) + 12*x^2*e^(1/5*x)*log( 5) - 4*x^3*log(1/5*x) + 6*x^2*e^(2/5*x)*log(1/5*x) + 12*x^2*e^(1/5*x)*log( 1/5*x) + 6*x^2*log(5) - 4*x*e^(3/5*x)*log(5) - 12*x*e^(2/5*x)*log(5) - 12* x*e^(1/5*x)*log(5) + 6*x^2*log(1/5*x) - 4*x*e^(3/5*x)*log(1/5*x) - 12*x*e^ (2/5*x)*log(1/5*x) - 12*x*e^(1/5*x)*log(1/5*x) + e^(4/5*x)*log(5) + 4*e^(3 /5*x)*log(5) + 6*e^(2/5*x)*log(5) + 4*e^(1/5*x)*log(5) + log(5)^2 - 4*x*lo g(1/5*x) + e^(4/5*x)*log(1/5*x) + 4*e^(3/5*x)*log(1/5*x) + 6*e^(2/5*x)*log (1/5*x) + 4*e^(1/5*x)*log(1/5*x) + 2*log(5)*log(1/5*x) + log(1/5*x)^2 + lo g(5) + log(1/5*x))/x
Timed out. \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\int \frac {2\,{\mathrm {e}}^{\frac {4\,x}{5}}-8\,x+\frac {{\mathrm {e}}^{\frac {2\,x}{5}}\,\left (60\,x^2-120\,x+60\right )}{5}-\frac {{\mathrm {e}}^{x/5}\,\left (40\,x^3-120\,x^2+120\,x-40\right )}{5}-2\,{\ln \left (x\right )}^2-\frac {{\mathrm {e}}^{\frac {3\,x}{5}}\,\left (40\,x-40\right )}{5}+12\,x^2-8\,x^3+2\,x^4+\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^{\frac {2\,x}{5}}\,\left (24\,x^3+12\,x^2+24\,x-60\right )-{\mathrm {e}}^{\frac {3\,x}{5}}\,\left (24\,x^2-24\,x+40\right )-{\mathrm {e}}^{x/5}\,\left (8\,x^4+56\,x^3-96\,x^2-8\,x+40\right )+{\mathrm {e}}^{\frac {4\,x}{5}}\,\left (8\,x-10\right )+60\,x^2-80\,x^3+30\,x^4+10\right )}{5}+2}{x^2} \,d x \] Input:
int((2*exp((4*x)/5) - 8*x + (exp((2*x)/5)*(60*x^2 - 120*x + 60))/5 - (exp( x/5)*(120*x - 120*x^2 + 40*x^3 - 40))/5 - 2*log(x)^2 - (exp((3*x)/5)*(40*x - 40))/5 + 12*x^2 - 8*x^3 + 2*x^4 + (log(x)*(exp((2*x)/5)*(24*x + 12*x^2 + 24*x^3 - 60) - exp((3*x)/5)*(24*x^2 - 24*x + 40) - exp(x/5)*(56*x^3 - 96 *x^2 - 8*x + 8*x^4 + 40) + exp((4*x)/5)*(8*x - 10) + 60*x^2 - 80*x^3 + 30* x^4 + 10))/5 + 2)/x^2,x)
Output:
int((2*exp((4*x)/5) - 8*x + (exp((2*x)/5)*(60*x^2 - 120*x + 60))/5 - (exp( x/5)*(120*x - 120*x^2 + 40*x^3 - 40))/5 - 2*log(x)^2 - (exp((3*x)/5)*(40*x - 40))/5 + 12*x^2 - 8*x^3 + 2*x^4 + (log(x)*(exp((2*x)/5)*(24*x + 12*x^2 + 24*x^3 - 60) - exp((3*x)/5)*(24*x^2 - 24*x + 40) - exp(x/5)*(56*x^3 - 96 *x^2 - 8*x + 8*x^4 + 40) + exp((4*x)/5)*(8*x - 10) + 60*x^2 - 80*x^3 + 30* x^4 + 10))/5 + 2)/x^2, x)
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.46 \[ \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{5 x^2} \, dx=\frac {2 \,\mathrm {log}\left (x \right ) \left (e^{\frac {4 x}{5}}-4 e^{\frac {3 x}{5}} x +4 e^{\frac {3 x}{5}}+6 e^{\frac {2 x}{5}} x^{2}-12 e^{\frac {2 x}{5}} x +6 e^{\frac {2 x}{5}}-4 e^{\frac {x}{5}} x^{3}+12 e^{\frac {x}{5}} x^{2}-12 e^{\frac {x}{5}} x +4 e^{\frac {x}{5}}+\mathrm {log}\left (x \right )+x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )}{x} \] Input:
int(1/5*(-10*log(x)^2+((8*x-10)*exp(1/5*x)^4+(-24*x^2+24*x-40)*exp(1/5*x)^ 3+(24*x^3+12*x^2+24*x-60)*exp(1/5*x)^2+(-8*x^4-56*x^3+96*x^2+8*x-40)*exp(1 /5*x)+30*x^4-80*x^3+60*x^2+10)*log(x)+10*exp(1/5*x)^4+(-40*x+40)*exp(1/5*x )^3+(60*x^2-120*x+60)*exp(1/5*x)^2+(-40*x^3+120*x^2-120*x+40)*exp(1/5*x)+1 0*x^4-40*x^3+60*x^2-40*x+10)/x^2,x)
Output:
(2*log(x)*(e**((4*x)/5) - 4*e**((3*x)/5)*x + 4*e**((3*x)/5) + 6*e**((2*x)/ 5)*x**2 - 12*e**((2*x)/5)*x + 6*e**((2*x)/5) - 4*e**(x/5)*x**3 + 12*e**(x/ 5)*x**2 - 12*e**(x/5)*x + 4*e**(x/5) + log(x) + x**4 - 4*x**3 + 6*x**2 - 4 *x + 1))/x