Integrand size = 107, antiderivative size = 330 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\frac {e^{-\frac {x}{y}} (3-4 \text {mc}) \text {mc}^8 \pi ^2}{384 x}+\frac {3}{8} e^{-\frac {x}{y}} \text {mc}^5 \pi ^2 y+\frac {1}{48} e^{-\frac {x}{y}} (3-22 \text {mc}) \text {mc}^2 \pi ^2 x y-\frac {1}{128} e^{-\frac {x}{y}} (1+4 \text {mc}) \pi ^2 x^2 y+\frac {1}{48} e^{-\frac {x}{y}} (3-22 \text {mc}) \text {mc}^2 \pi ^2 y^2+\frac {1}{4} e^{-\frac {x}{y}} \text {mc}^3 \pi ^2 y^2-\frac {1}{64} e^{-\frac {x}{y}} (1+4 \text {mc}) \pi ^2 x y^2-\frac {1}{64} e^{-\frac {x}{y}} (1+4 \text {mc}) \pi ^2 y^3+\frac {1}{16} (1-2 \text {mc}) \text {mc}^6 \pi ^2 \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )+\frac {(3-4 \text {mc}) \text {mc}^8 \pi ^2 \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )}{384 y}+\frac {1}{32} \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 y\right ) y \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )-\frac {1}{32} e^{-\frac {x}{y}} \text {mc}^3 \pi ^2 (3 (1-4 \text {mc}) \text {mc}-8 x) y \log \left (\frac {x}{\text {mc}^2}\right )+\frac {1}{4} e^{-\frac {x}{y}} \text {mc}^3 \pi ^2 y^2 \log \left (\frac {x}{\text {mc}^2}\right ) \]
1/384*(3-4*mc)*mc^8*Pi^2/exp(x/y)/x+3/8*mc^5*Pi^2*y/exp(x/y)+1/48*(3-22*mc )*mc^2*Pi^2*x*y/exp(x/y)-1/128*(1+4*mc)*Pi^2*x^2*y/exp(x/y)+1/48*(3-22*mc) *mc^2*Pi^2*y^2/exp(x/y)+1/4*mc^3*Pi^2*y^2/exp(x/y)-1/64*(1+4*mc)*Pi^2*x*y^ 2/exp(x/y)-1/64*(1+4*mc)*Pi^2*y^3/exp(x/y)+1/16*(1-2*mc)*mc^6*Pi^2*Ei(-x/y )+1/384*(3-4*mc)*mc^8*Pi^2*Ei(-x/y)/y+1/32*mc^3*Pi^2*(-12*mc^2+3*mc-8*y)*y *Ei(-x/y)-1/32*mc^3*Pi^2*(3*(1-4*mc)*mc-8*x)*y*ln(x/mc^2)/exp(x/y)+1/4*mc^ 3*Pi^2*y^2*ln(x/mc^2)/exp(x/y)
Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.55 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\frac {1}{384} \pi ^2 \left (-\frac {\text {mc}^3 \left (-3 \text {mc}^5+4 \text {mc}^6-24 \text {mc}^3 y+48 \text {mc}^4 y-36 \text {mc} y^2+144 \text {mc}^2 y^2+96 y^3\right ) \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )}{y}+\frac {e^{-\frac {x}{y}} \left (3 \text {mc}^8-4 \text {mc}^9+144 \text {mc}^5 x y+24 \text {mc}^2 x y (x+y)-16 \text {mc}^3 x y (11 x+5 y)-3 x y \left (x^2+2 x y+2 y^2\right )-12 \text {mc} x y \left (x^2+2 x y+2 y^2\right )+12 \text {mc}^3 x y \left (-3 \text {mc}+12 \text {mc}^2+8 (x+y)\right ) \log \left (\frac {x}{\text {mc}^2}\right )\right )}{x}\right ) \]
Integrate[(Pi^2*(-3*mc^8 + 4*mc^9 + 24*mc^6*x - 48*mc^7*x - 144*mc^5*x^2 - 24*mc^2*x^3 + 176*mc^3*x^3 + 3*x^4 + 12*mc*x^4) + 12*mc^3*Pi^2*(3*mc - 12 *mc^2 - 8*x)*x^2*Log[x/mc^2])/(384*E^(x/y)*x^2),x]
(Pi^2*(-((mc^3*(-3*mc^5 + 4*mc^6 - 24*mc^3*y + 48*mc^4*y - 36*mc*y^2 + 144 *mc^2*y^2 + 96*y^3)*ExpIntegralEi[-(x/y)])/y) + (3*mc^8 - 4*mc^9 + 144*mc^ 5*x*y + 24*mc^2*x*y*(x + y) - 16*mc^3*x*y*(11*x + 5*y) - 3*x*y*(x^2 + 2*x* y + 2*y^2) - 12*mc*x*y*(x^2 + 2*x*y + 2*y^2) + 12*mc^3*x*y*(-3*mc + 12*mc^ 2 + 8*(x + y))*Log[x/mc^2])/(E^(x/y)*x)))/384
Time = 1.03 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {27, 25, 7293, 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.
| \(\displaystyle \int \frac {e^{-\frac {x}{y}} \left (12 \pi ^2 \text {mc}^3 x^2 \left (-12 \text {mc}^2+3 \text {mc}-8 x\right ) \log \left (\frac {x}{\text {mc}^2}\right )+\pi ^2 \left (4 \text {mc}^9-3 \text {mc}^8-48 \text {mc}^7 x+24 \text {mc}^6 x-144 \text {mc}^5 x^2+176 \text {mc}^3 x^3-24 \text {mc}^2 x^3+12 \text {mc} x^4+3 x^4\right )\right )}{384 x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{384} \int -\frac {e^{-\frac {x}{y}} \left (\pi ^2 \left ((3-4 \text {mc}) \text {mc}^8+48 x \text {mc}^7-24 x \text {mc}^6+144 x^2 \text {mc}^5-176 x^3 \text {mc}^3+24 x^3 \text {mc}^2-12 x^4 \text {mc}-3 x^4\right )-12 \text {mc}^3 \pi ^2 (3 (1-4 \text {mc}) \text {mc}-8 x) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{384} \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left ((3-4 \text {mc}) \text {mc}^8+48 x \text {mc}^7-24 x \text {mc}^6+144 x^2 \text {mc}^5-176 x^3 \text {mc}^3+24 x^3 \text {mc}^2-12 x^4 \text {mc}-3 x^4\right )-12 \text {mc}^3 \pi ^2 (3 (1-4 \text {mc}) \text {mc}-8 x) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{384} \int \left (12 e^{-\frac {x}{y}} \pi ^2 \left (12 \text {mc}^2-3 \text {mc}+8 x\right ) \log \left (\frac {x}{\text {mc}^2}\right ) \text {mc}^3+\frac {e^{-\frac {x}{y}} \pi ^2 \left (\text {mc}^2-x\right ) \left ((3-4 \text {mc}) \text {mc}^6-(21-44 \text {mc}) x \text {mc}^4-(21-188 \text {mc}) x^2 \text {mc}^2+3 (4 \text {mc}+1) x^3\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{384} \left (\frac {\pi ^2 (3-4 \text {mc}) \text {mc}^8 \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )}{y}+24 \pi ^2 (1-2 \text {mc}) \text {mc}^6 \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )+12 \pi ^2 \text {mc}^3 y \left (-12 \text {mc}^2+3 \text {mc}-8 y\right ) \operatorname {ExpIntegralEi}\left (-\frac {x}{y}\right )+\frac {\pi ^2 (3-4 \text {mc}) \text {mc}^8 e^{-\frac {x}{y}}}{x}+144 \pi ^2 \text {mc}^5 y e^{-\frac {x}{y}}+96 \pi ^2 \text {mc}^3 y^2 e^{-\frac {x}{y}}+8 \pi ^2 (3-22 \text {mc}) \text {mc}^2 y^2 e^{-\frac {x}{y}}+8 \pi ^2 (3-22 \text {mc}) \text {mc}^2 x y e^{-\frac {x}{y}}+96 \pi ^2 \text {mc}^3 y^2 e^{-\frac {x}{y}} \log \left (\frac {x}{\text {mc}^2}\right )-12 \pi ^2 \text {mc}^3 y (3 (1-4 \text {mc}) \text {mc}-8 x) e^{-\frac {x}{y}} \log \left (\frac {x}{\text {mc}^2}\right )-3 \pi ^2 (4 \text {mc}+1) x^2 y e^{-\frac {x}{y}}-6 \pi ^2 (4 \text {mc}+1) y^3 e^{-\frac {x}{y}}-6 \pi ^2 (4 \text {mc}+1) x y^2 e^{-\frac {x}{y}}\right )\) |
Int[(Pi^2*(-3*mc^8 + 4*mc^9 + 24*mc^6*x - 48*mc^7*x - 144*mc^5*x^2 - 24*mc ^2*x^3 + 176*mc^3*x^3 + 3*x^4 + 12*mc*x^4) + 12*mc^3*Pi^2*(3*mc - 12*mc^2 - 8*x)*x^2*Log[x/mc^2])/(384*E^(x/y)*x^2),x]
(((3 - 4*mc)*mc^8*Pi^2)/(E^(x/y)*x) + (144*mc^5*Pi^2*y)/E^(x/y) + (8*(3 - 22*mc)*mc^2*Pi^2*x*y)/E^(x/y) - (3*(1 + 4*mc)*Pi^2*x^2*y)/E^(x/y) + (8*(3 - 22*mc)*mc^2*Pi^2*y^2)/E^(x/y) + (96*mc^3*Pi^2*y^2)/E^(x/y) - (6*(1 + 4*m c)*Pi^2*x*y^2)/E^(x/y) - (6*(1 + 4*mc)*Pi^2*y^3)/E^(x/y) + 24*(1 - 2*mc)*m c^6*Pi^2*ExpIntegralEi[-(x/y)] + ((3 - 4*mc)*mc^8*Pi^2*ExpIntegralEi[-(x/y )])/y + 12*mc^3*Pi^2*(3*mc - 12*mc^2 - 8*y)*y*ExpIntegralEi[-(x/y)] - (12* mc^3*Pi^2*(3*(1 - 4*mc)*mc - 8*x)*y*Log[x/mc^2])/E^(x/y) + (96*mc^3*Pi^2*y ^2*Log[x/mc^2])/E^(x/y))/384
3.3.82.3.1 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]]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.70 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.11
int(1/384*(Pi^2*(4*mc^9-3*mc^8-48*mc^7*x+24*mc^6*x-144*mc^5*x^2+176*mc^3*x ^3-24*mc^2*x^3+12*mc*x^4+3*x^4)+12*mc^3*Pi^2*(-12*mc^2+3*mc-8*x)*x^2*ln(x/ mc^2))/exp(x/y)/x^2,x,method=_RETURNVERBOSE)
1/8*I*y*Pi^3*mc^3*csgn(I/mc^2*x)^2*csgn(I*x)*exp(-x/y)*x+1/8*I*y*Pi^3*mc^3 *csgn(I/mc^2)*csgn(I/mc^2*x)^2*exp(-x/y)*x-1/4*I*y*Pi^3*mc^3*csgn(I*mc)*cs gn(I*mc^2)^2*exp(-x/y)*x+1/8*I*y*Pi^3*mc^3*csgn(I*mc)^2*csgn(I*mc^2)*exp(- x/y)*x+3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2)*csgn(I/mc^2*x)*csgn(I*x)- 3/16*I*y*Pi^3*exp(-x/y)*mc^5*csgn(I/mc^2)*csgn(I/mc^2*x)*csgn(I*x)-1/8*I*y ^2*Pi^3*mc^3*csgn(I/mc^2)*csgn(I/mc^2*x)*csgn(I*x)*exp(-x/y)-3/64*I*y*Pi^3 *exp(-x/y)*mc^4*csgn(I*mc)^2*csgn(I*mc^2)+3/32*I*y*Pi^3*exp(-x/y)*mc^4*csg n(I*mc)*csgn(I*mc^2)^2-3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2*x)^2*csgn( I*x)+3/16*I*y*Pi^3*exp(-x/y)*mc^5*csgn(I/mc^2)*csgn(I/mc^2*x)^2+1/8*I*y*Pi ^3*mc^3*csgn(I*mc^2)^3*exp(-x/y)*x-1/8*I*y*Pi^3*mc^3*csgn(I/mc^2*x)^3*exp( -x/y)*x-1/4*I*y^2*Pi^3*mc^3*csgn(I*mc)*csgn(I*mc^2)^2*exp(-x/y)+1/8*I*y^2* Pi^3*mc^3*csgn(I*mc)^2*csgn(I*mc^2)*exp(-x/y)+1/8*I*y^2*Pi^3*mc^3*csgn(I/m c^2*x)^2*csgn(I*x)*exp(-x/y)+1/8*I*y^2*Pi^3*mc^3*csgn(I/mc^2)*csgn(I/mc^2* x)^2*exp(-x/y)+3/16*I*y*Pi^3*exp(-x/y)*mc^5*csgn(I*mc)^2*csgn(I*mc^2)-3/8* I*y*Pi^3*exp(-x/y)*mc^5*csgn(I*mc)*csgn(I*mc^2)^2+3/16*I*y*Pi^3*exp(-x/y)* mc^5*csgn(I/mc^2*x)^2*csgn(I*x)-3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2)* csgn(I/mc^2*x)^2-1/8*I*y*Pi^3*mc^3*csgn(I/mc^2)*csgn(I/mc^2*x)*csgn(I*x)*e xp(-x/y)*x-1/128/y*Pi^2*mc^8*Ei(1,x/y)+1/96/y*Pi^2*mc^9*Ei(1,x/y)-1/128*y* Pi^2*exp(-x/y)*x^2-1/64*y^2*Pi^2*x*exp(-x/y)-1/16*y^3*Pi^2*mc*exp(-x/y)+1/ 16*y^2*Pi^2*mc^2*exp(-x/y)+3/8*y*Pi^2*exp(-x/y)*mc^5-1/96*Pi^2*mc^9/x*e...
Time = 0.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\frac {12 \, {\left (8 \, \pi ^{2} \mathit {mc}^{3} x y^{3} + {\left (8 \, \pi ^{2} \mathit {mc}^{3} x^{2} + 3 \, \pi ^{2} {\left (4 \, \mathit {mc}^{5} - \mathit {mc}^{4}\right )} x\right )} y^{2}\right )} e^{\left (-\frac {x}{y}\right )} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - {\left (96 \, \pi ^{2} \mathit {mc}^{3} x y^{3} + 36 \, \pi ^{2} {\left (4 \, \mathit {mc}^{5} - \mathit {mc}^{4}\right )} x y^{2} + 24 \, \pi ^{2} {\left (2 \, \mathit {mc}^{7} - \mathit {mc}^{6}\right )} x y + \pi ^{2} {\left (4 \, \mathit {mc}^{9} - 3 \, \mathit {mc}^{8}\right )} x\right )} {\rm Ei}\left (-\frac {x}{y}\right ) - {\left (6 \, \pi ^{2} {\left (4 \, \mathit {mc} + 1\right )} x y^{4} + \pi ^{2} {\left (4 \, \mathit {mc}^{9} - 3 \, \mathit {mc}^{8}\right )} y + 2 \, {\left (3 \, \pi ^{2} {\left (4 \, \mathit {mc} + 1\right )} x^{2} + 4 \, \pi ^{2} {\left (10 \, \mathit {mc}^{3} - 3 \, \mathit {mc}^{2}\right )} x\right )} y^{3} - {\left (144 \, \pi ^{2} \mathit {mc}^{5} x - 3 \, \pi ^{2} {\left (4 \, \mathit {mc} + 1\right )} x^{3} - 8 \, \pi ^{2} {\left (22 \, \mathit {mc}^{3} - 3 \, \mathit {mc}^{2}\right )} x^{2}\right )} y^{2}\right )} e^{\left (-\frac {x}{y}\right )}}{384 \, x y} \]
integrate(1/384*(pi^2*(4*mc^9-3*mc^8-48*mc^7*x+24*mc^6*x-144*mc^5*x^2+176* mc^3*x^3-24*mc^2*x^3+12*mc*x^4+3*x^4)+12*mc^3*pi^2*(-12*mc^2+3*mc-8*x)*x^2 *log(x/mc^2))/exp(x/y)/x^2,x, algorithm="fricas")
1/384*(12*(8*pi^2*mc^3*x*y^3 + (8*pi^2*mc^3*x^2 + 3*pi^2*(4*mc^5 - mc^4)*x )*y^2)*e^(-x/y)*log(x/mc^2) - (96*pi^2*mc^3*x*y^3 + 36*pi^2*(4*mc^5 - mc^4 )*x*y^2 + 24*pi^2*(2*mc^7 - mc^6)*x*y + pi^2*(4*mc^9 - 3*mc^8)*x)*Ei(-x/y) - (6*pi^2*(4*mc + 1)*x*y^4 + pi^2*(4*mc^9 - 3*mc^8)*y + 2*(3*pi^2*(4*mc + 1)*x^2 + 4*pi^2*(10*mc^3 - 3*mc^2)*x)*y^3 - (144*pi^2*mc^5*x - 3*pi^2*(4* mc + 1)*x^3 - 8*pi^2*(22*mc^3 - 3*mc^2)*x^2)*y^2)*e^(-x/y))/(x*y)
Time = 6.17 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=- \frac {\pi ^{2} mc^{9} \operatorname {E}_{2}\left (\frac {x}{y}\right )}{96 x} + \frac {\pi ^{2} mc^{8} \operatorname {E}_{2}\left (\frac {x}{y}\right )}{128 x} - \frac {\pi ^{2} mc^{7} \operatorname {Ei}{\left (- \frac {x}{y} \right )}}{8} + \frac {\pi ^{2} mc^{6} \operatorname {Ei}{\left (- \frac {x}{y} \right )}}{16} + \frac {3 \pi ^{2} mc^{5} y e^{- \frac {x}{y}}}{8} - \frac {3 \pi ^{2} mc^{5} \left (y \operatorname {Ei}{\left (- \frac {x}{y} \right )} - y e^{- \frac {x}{y}} \log {\left (\frac {x}{mc^{2}} \right )}\right )}{8} + \frac {3 \pi ^{2} mc^{4} \left (y \operatorname {Ei}{\left (- \frac {x}{y} \right )} - y e^{- \frac {x}{y}} \log {\left (\frac {x}{mc^{2}} \right )}\right )}{32} + \frac {11 \pi ^{2} mc^{3} \left (- x y e^{- \frac {x}{y}} - y^{2} e^{- \frac {x}{y}}\right )}{24} - \frac {\pi ^{2} mc^{3} \left (y^{2} \operatorname {Ei}{\left (- \frac {x}{y} \right )} - y^{2} e^{- \frac {x}{y}} + \left (- x y e^{- \frac {x}{y}} - y^{2} e^{- \frac {x}{y}}\right ) \log {\left (\frac {x}{mc^{2}} \right )}\right )}{4} - \frac {\pi ^{2} mc^{2} \left (- x y e^{- \frac {x}{y}} - y^{2} e^{- \frac {x}{y}}\right )}{16} + \frac {\pi ^{2} mc \left (- x^{2} y e^{- \frac {x}{y}} - 2 x y^{2} e^{- \frac {x}{y}} - 2 y^{3} e^{- \frac {x}{y}}\right )}{32} + \frac {\pi ^{2} \left (- x^{2} y e^{- \frac {x}{y}} - 2 x y^{2} e^{- \frac {x}{y}} - 2 y^{3} e^{- \frac {x}{y}}\right )}{128} \]
integrate(1/384*(pi**2*(4*mc**9-3*mc**8-48*mc**7*x+24*mc**6*x-144*mc**5*x* *2+176*mc**3*x**3-24*mc**2*x**3+12*mc*x**4+3*x**4)+12*mc**3*pi**2*(-12*mc* *2+3*mc-8*x)*x**2*ln(x/mc**2))/exp(x/y)/x**2,x)
-pi**2*mc**9*expint(2, x/y)/(96*x) + pi**2*mc**8*expint(2, x/y)/(128*x) - pi**2*mc**7*Ei(-x/y)/8 + pi**2*mc**6*Ei(-x/y)/16 + 3*pi**2*mc**5*y*exp(-x/ y)/8 - 3*pi**2*mc**5*(y*Ei(-x/y) - y*exp(-x/y)*log(x/mc**2))/8 + 3*pi**2*m c**4*(y*Ei(-x/y) - y*exp(-x/y)*log(x/mc**2))/32 + 11*pi**2*mc**3*(-x*y*exp (-x/y) - y**2*exp(-x/y))/24 - pi**2*mc**3*(y**2*Ei(-x/y) - y**2*exp(-x/y) + (-x*y*exp(-x/y) - y**2*exp(-x/y))*log(x/mc**2))/4 - pi**2*mc**2*(-x*y*ex p(-x/y) - y**2*exp(-x/y))/16 + pi**2*mc*(-x**2*y*exp(-x/y) - 2*x*y**2*exp( -x/y) - 2*y**3*exp(-x/y))/32 + pi**2*(-x**2*y*exp(-x/y) - 2*x*y**2*exp(-x/ y) - 2*y**3*exp(-x/y))/128
\[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\int { -\frac {{\left (12 \, \pi ^{2} {\left (12 \, \mathit {mc}^{2} - 3 \, \mathit {mc} + 8 \, x\right )} \mathit {mc}^{3} x^{2} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - \pi ^{2} {\left (4 \, \mathit {mc}^{9} - 3 \, \mathit {mc}^{8} - 48 \, \mathit {mc}^{7} x + 24 \, \mathit {mc}^{6} x - 144 \, \mathit {mc}^{5} x^{2} + 176 \, \mathit {mc}^{3} x^{3} - 24 \, \mathit {mc}^{2} x^{3} + 12 \, \mathit {mc} x^{4} + 3 \, x^{4}\right )}\right )} e^{\left (-\frac {x}{y}\right )}}{384 \, x^{2}} \,d x } \]
integrate(1/384*(pi^2*(4*mc^9-3*mc^8-48*mc^7*x+24*mc^6*x-144*mc^5*x^2+176* mc^3*x^3-24*mc^2*x^3+12*mc*x^4+3*x^4)+12*mc^3*pi^2*(-12*mc^2+3*mc-8*x)*x^2 *log(x/mc^2))/exp(x/y)/x^2,x, algorithm="maxima")
-1/96*pi^2*mc^9*gamma(-1, x/y)/y - 1/8*pi^2*mc^7*Ei(-x/y) + 1/128*pi^2*mc^ 8*gamma(-1, x/y)/y + 3/8*pi^2*mc^5*y*e^(-x/y)*log(x/mc^2) + 1/16*pi^2*mc^6 *Ei(-x/y) - 3/8*pi^2*mc^5*y*Ei(-x/y) + 3/8*pi^2*mc^5*y*e^(-x/y) - 3/32*pi^ 2*mc^4*y*e^(-x/y)*log(x/mc^2) + 3/32*pi^2*mc^4*y*Ei(-x/y) - 11/24*pi^2*(x* y + y^2)*mc^3*e^(-x/y) + 1/4*pi^2*((x*y + y^2)*e^(-x/y)*log(x) + integrate ((2*x^2*log(mc) - x*y - y^2)*e^(-x/y)/x, x))*mc^3 + 1/16*pi^2*(x*y + y^2)* mc^2*e^(-x/y) - 1/32*pi^2*(x^2*y + 2*x*y^2 + 2*y^3)*mc*e^(-x/y) - 1/128*pi ^2*(x^2*y + 2*x*y^2 + 2*y^3)*e^(-x/y)
Time = 0.29 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=-\frac {4 \, \pi ^{2} \mathit {mc}^{9} x {\rm Ei}\left (-\frac {x}{y}\right ) + 4 \, \pi ^{2} \mathit {mc}^{9} y e^{\left (-\frac {x}{y}\right )} - 3 \, \pi ^{2} \mathit {mc}^{8} x {\rm Ei}\left (-\frac {x}{y}\right ) + 48 \, \pi ^{2} \mathit {mc}^{7} x y {\rm Ei}\left (-\frac {x}{y}\right ) - 3 \, \pi ^{2} \mathit {mc}^{8} y e^{\left (-\frac {x}{y}\right )} - 144 \, \pi ^{2} \mathit {mc}^{5} x y^{2} e^{\left (-\frac {x}{y}\right )} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - 24 \, \pi ^{2} \mathit {mc}^{6} x y {\rm Ei}\left (-\frac {x}{y}\right ) + 144 \, \pi ^{2} \mathit {mc}^{5} x y^{2} {\rm Ei}\left (-\frac {x}{y}\right ) - 144 \, \pi ^{2} \mathit {mc}^{5} x y^{2} e^{\left (-\frac {x}{y}\right )} + 36 \, \pi ^{2} \mathit {mc}^{4} x y^{2} e^{\left (-\frac {x}{y}\right )} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - 96 \, \pi ^{2} \mathit {mc}^{3} x^{2} y^{2} e^{\left (-\frac {x}{y}\right )} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - 96 \, \pi ^{2} \mathit {mc}^{3} x y^{3} e^{\left (-\frac {x}{y}\right )} \log \left (\frac {x}{\mathit {mc}^{2}}\right ) - 36 \, \pi ^{2} \mathit {mc}^{4} x y^{2} {\rm Ei}\left (-\frac {x}{y}\right ) + 96 \, \pi ^{2} \mathit {mc}^{3} x y^{3} {\rm Ei}\left (-\frac {x}{y}\right ) + 176 \, \pi ^{2} \mathit {mc}^{3} x^{2} y^{2} e^{\left (-\frac {x}{y}\right )} + 80 \, \pi ^{2} \mathit {mc}^{3} x y^{3} e^{\left (-\frac {x}{y}\right )} - 24 \, \pi ^{2} \mathit {mc}^{2} x^{2} y^{2} e^{\left (-\frac {x}{y}\right )} + 12 \, \pi ^{2} \mathit {mc} x^{3} y^{2} e^{\left (-\frac {x}{y}\right )} - 24 \, \pi ^{2} \mathit {mc}^{2} x y^{3} e^{\left (-\frac {x}{y}\right )} + 24 \, \pi ^{2} \mathit {mc} x^{2} y^{3} e^{\left (-\frac {x}{y}\right )} + 24 \, \pi ^{2} \mathit {mc} x y^{4} e^{\left (-\frac {x}{y}\right )} + 3 \, \pi ^{2} x^{3} y^{2} e^{\left (-\frac {x}{y}\right )} + 6 \, \pi ^{2} x^{2} y^{3} e^{\left (-\frac {x}{y}\right )} + 6 \, \pi ^{2} x y^{4} e^{\left (-\frac {x}{y}\right )}}{384 \, x y} \]
integrate(1/384*(pi^2*(4*mc^9-3*mc^8-48*mc^7*x+24*mc^6*x-144*mc^5*x^2+176* mc^3*x^3-24*mc^2*x^3+12*mc*x^4+3*x^4)+12*mc^3*pi^2*(-12*mc^2+3*mc-8*x)*x^2 *log(x/mc^2))/exp(x/y)/x^2,x, algorithm="giac")
-1/384*(4*pi^2*mc^9*x*Ei(-x/y) + 4*pi^2*mc^9*y*e^(-x/y) - 3*pi^2*mc^8*x*Ei (-x/y) + 48*pi^2*mc^7*x*y*Ei(-x/y) - 3*pi^2*mc^8*y*e^(-x/y) - 144*pi^2*mc^ 5*x*y^2*e^(-x/y)*log(x/mc^2) - 24*pi^2*mc^6*x*y*Ei(-x/y) + 144*pi^2*mc^5*x *y^2*Ei(-x/y) - 144*pi^2*mc^5*x*y^2*e^(-x/y) + 36*pi^2*mc^4*x*y^2*e^(-x/y) *log(x/mc^2) - 96*pi^2*mc^3*x^2*y^2*e^(-x/y)*log(x/mc^2) - 96*pi^2*mc^3*x* y^3*e^(-x/y)*log(x/mc^2) - 36*pi^2*mc^4*x*y^2*Ei(-x/y) + 96*pi^2*mc^3*x*y^ 3*Ei(-x/y) + 176*pi^2*mc^3*x^2*y^2*e^(-x/y) + 80*pi^2*mc^3*x*y^3*e^(-x/y) - 24*pi^2*mc^2*x^2*y^2*e^(-x/y) + 12*pi^2*mc*x^3*y^2*e^(-x/y) - 24*pi^2*mc ^2*x*y^3*e^(-x/y) + 24*pi^2*mc*x^2*y^3*e^(-x/y) + 24*pi^2*mc*x*y^4*e^(-x/y ) + 3*pi^2*x^3*y^2*e^(-x/y) + 6*pi^2*x^2*y^3*e^(-x/y) + 6*pi^2*x*y^4*e^(-x /y))/(x*y)
Time = 0.76 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\mathrm {ei}\left (-\frac {x}{y}\right )\,\left (\frac {\frac {\Pi ^2\,{\mathrm {mc}}^8}{128}-\frac {\Pi ^2\,{\mathrm {mc}}^9}{96}}{y}+\frac {\Pi ^2\,{\mathrm {mc}}^6}{16}-\frac {\Pi ^2\,{\mathrm {mc}}^7}{8}+y\,\left (\frac {3\,\Pi ^2\,{\mathrm {mc}}^4}{32}-\frac {3\,\Pi ^2\,{\mathrm {mc}}^5}{8}\right )-\frac {\Pi ^2\,{\mathrm {mc}}^3\,y^2}{4}\right )-\frac {2\,\Pi ^2\,x^2\,y\,{\mathrm {e}}^{-\frac {x}{y}}\,\left (-72\,{\mathrm {mc}}^5+40\,{\mathrm {mc}}^3\,y-12\,{\mathrm {mc}}^2\,y+12\,\mathrm {mc}\,y^2+3\,y^2\right )+2\,\Pi ^2\,x^3\,y\,{\mathrm {e}}^{-\frac {x}{y}}\,\left (88\,{\mathrm {mc}}^3-12\,{\mathrm {mc}}^2+12\,y\,\mathrm {mc}+3\,y\right )+\Pi ^2\,{\mathrm {mc}}^8\,x\,{\mathrm {e}}^{-\frac {x}{y}}\,\left (4\,\mathrm {mc}-3\right )+3\,\Pi ^2\,x^4\,y\,{\mathrm {e}}^{-\frac {x}{y}}\,\left (4\,\mathrm {mc}+1\right )-96\,\Pi ^2\,{\mathrm {mc}}^3\,x^3\,y\,\ln \left (\frac {x}{{\mathrm {mc}}^2}\right )\,{\mathrm {e}}^{-\frac {x}{y}}-12\,\Pi ^2\,{\mathrm {mc}}^3\,x^2\,y\,\ln \left (\frac {x}{{\mathrm {mc}}^2}\right )\,{\mathrm {e}}^{-\frac {x}{y}}\,\left (12\,{\mathrm {mc}}^2-3\,\mathrm {mc}+8\,y\right )}{384\,x^2} \]
int((exp(-x/y)*((Pi^2*(176*mc^3*x^3 - 24*mc^2*x^3 - 144*mc^5*x^2 + 12*mc*x ^4 + 24*mc^6*x - 48*mc^7*x - 3*mc^8 + 4*mc^9 + 3*x^4))/384 - (Pi^2*mc^3*x^ 2*log(x/mc^2)*(8*x - 3*mc + 12*mc^2))/32))/x^2,x)
ei(-x/y)*(((Pi^2*mc^8)/128 - (Pi^2*mc^9)/96)/y + (Pi^2*mc^6)/16 - (Pi^2*mc ^7)/8 + y*((3*Pi^2*mc^4)/32 - (3*Pi^2*mc^5)/8) - (Pi^2*mc^3*y^2)/4) - (2*P i^2*x^2*y*exp(-x/y)*(12*mc*y^2 - 12*mc^2*y + 40*mc^3*y - 72*mc^5 + 3*y^2) + 2*Pi^2*x^3*y*exp(-x/y)*(3*y + 12*mc*y - 12*mc^2 + 88*mc^3) + Pi^2*mc^8*x *exp(-x/y)*(4*mc - 3) + 3*Pi^2*x^4*y*exp(-x/y)*(4*mc + 1) - 96*Pi^2*mc^3*x ^3*y*log(x/mc^2)*exp(-x/y) - 12*Pi^2*mc^3*x^2*y*log(x/mc^2)*exp(-x/y)*(8*y - 3*mc + 12*mc^2))/(384*x^2)
Time = 0.00 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\frac {x}{y}} \left (\pi ^2 \left (-3 \text {mc}^8+4 \text {mc}^9+24 \text {mc}^6 x-48 \text {mc}^7 x-144 \text {mc}^5 x^2-24 \text {mc}^2 x^3+176 \text {mc}^3 x^3+3 x^4+12 \text {mc} x^4\right )+12 \text {mc}^3 \pi ^2 \left (3 \text {mc}-12 \text {mc}^2-8 x\right ) x^2 \log \left (\frac {x}{\text {mc}^2}\right )\right )}{384 x^2} \, dx=\frac {\pi ^{2} \left (-4 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{9} x +3 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{8} x -48 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{7} x y +24 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{6} x y -144 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{5} x \,y^{2}+36 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{4} x \,y^{2}-96 e^{\frac {x}{y}} \mathit {ei} \left (-\frac {x}{y}\right ) \mathit {mc}^{3} x \,y^{3}+144 \,\mathrm {log}\left (\frac {x}{\mathit {mc}^{2}}\right ) \mathit {mc}^{5} x \,y^{2}-36 \,\mathrm {log}\left (\frac {x}{\mathit {mc}^{2}}\right ) \mathit {mc}^{4} x \,y^{2}+96 \,\mathrm {log}\left (\frac {x}{\mathit {mc}^{2}}\right ) \mathit {mc}^{3} x^{2} y^{2}+96 \,\mathrm {log}\left (\frac {x}{\mathit {mc}^{2}}\right ) \mathit {mc}^{3} x \,y^{3}-4 \mathit {mc}^{9} y +3 \mathit {mc}^{8} y +144 \mathit {mc}^{5} x \,y^{2}-176 \mathit {mc}^{3} x^{2} y^{2}-80 \mathit {mc}^{3} x \,y^{3}+24 \mathit {mc}^{2} x^{2} y^{2}+24 \mathit {mc}^{2} x \,y^{3}-12 \mathit {mc} \,x^{3} y^{2}-24 \mathit {mc} \,x^{2} y^{3}-24 \mathit {mc} x \,y^{4}-3 x^{3} y^{2}-6 x^{2} y^{3}-6 x \,y^{4}\right )}{384 e^{\frac {x}{y}} x y} \]
int((pi**2*( - 144*log(x/mc**2)*mc**5*x**2 + 36*log(x/mc**2)*mc**4*x**2 - 96*log(x/mc**2)*mc**3*x**3 + 4*mc**9 - 3*mc**8 - 48*mc**7*x + 24*mc**6*x - 144*mc**5*x**2 + 176*mc**3*x**3 - 24*mc**2*x**3 + 12*mc*x**4 + 3*x**4))/( 384*e**(x/y)*x**2),x)
(pi**2*( - 4*e**(x/y)*ei(( - x)/y)*mc**9*x + 3*e**(x/y)*ei(( - x)/y)*mc**8 *x - 48*e**(x/y)*ei(( - x)/y)*mc**7*x*y + 24*e**(x/y)*ei(( - x)/y)*mc**6*x *y - 144*e**(x/y)*ei(( - x)/y)*mc**5*x*y**2 + 36*e**(x/y)*ei(( - x)/y)*mc* *4*x*y**2 - 96*e**(x/y)*ei(( - x)/y)*mc**3*x*y**3 + 144*log(x/mc**2)*mc**5 *x*y**2 - 36*log(x/mc**2)*mc**4*x*y**2 + 96*log(x/mc**2)*mc**3*x**2*y**2 + 96*log(x/mc**2)*mc**3*x*y**3 - 4*mc**9*y + 3*mc**8*y + 144*mc**5*x*y**2 - 176*mc**3*x**2*y**2 - 80*mc**3*x*y**3 + 24*mc**2*x**2*y**2 + 24*mc**2*x*y **3 - 12*mc*x**3*y**2 - 24*mc*x**2*y**3 - 24*mc*x*y**4 - 3*x**3*y**2 - 6*x **2*y**3 - 6*x*y**4))/(384*e**(x/y)*x*y)