3.282 \(\int \frac{e^{-\frac{x}{y}} (\pi ^2 (-3 \text{mc}^8+4 \text{mc}^9+24 \text{mc}^6 x-48 \text{mc}^7 x-144 \text{mc}^5 x^2-24 \text{mc}^2 x^3+176 \text{mc}^3 x^3+3 x^4+12 \text{mc} x^4)+12 \text{mc}^3 \pi ^2 (3 \text{mc}-12 \text{mc}^2-8 x) x^2 \log (\frac{x}{\text{mc}^2}))}{384 x^2} \, dx\)

Optimal. Leaf size=330 \[ \frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )}{384 y}+\frac{1}{16} \pi ^2 (1-2 \text{mc}) \text{mc}^6 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{1}{32} \pi ^2 \text{mc}^3 y \left (-12 \text{mc}^2+3 \text{mc}-8 y\right ) \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 y^2 e^{-\frac{x}{y}}+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )+\frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 e^{-\frac{x}{y}}}{384 x}+\frac{3}{8} \pi ^2 \text{mc}^5 y e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 x y e^{-\frac{x}{y}}-\frac{1}{32} \pi ^2 \text{mc}^3 y (3 (1-4 \text{mc}) \text{mc}-8 x) e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )-\frac{1}{128} \pi ^2 (4 \text{mc}+1) x^2 y e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) y^3 e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) x y^2 e^{-\frac{x}{y}} \]

[Out]

((3 - 4*mc)*mc^8*Pi^2)/(384*E^(x/y)*x) + (3*mc^5*Pi^2*y)/(8*E^(x/y)) + ((3 - 22*mc)*mc^2*Pi^2*x*y)/(48*E^(x/y)
) - ((1 + 4*mc)*Pi^2*x^2*y)/(128*E^(x/y)) + ((3 - 22*mc)*mc^2*Pi^2*y^2)/(48*E^(x/y)) + (mc^3*Pi^2*y^2)/(4*E^(x
/y)) - ((1 + 4*mc)*Pi^2*x*y^2)/(64*E^(x/y)) - ((1 + 4*mc)*Pi^2*y^3)/(64*E^(x/y)) + ((1 - 2*mc)*mc^6*Pi^2*ExpIn
tegralEi[-(x/y)])/16 + ((3 - 4*mc)*mc^8*Pi^2*ExpIntegralEi[-(x/y)])/(384*y) + (mc^3*Pi^2*(3*mc - 12*mc^2 - 8*y
)*y*ExpIntegralEi[-(x/y)])/32 - (mc^3*Pi^2*(3*(1 - 4*mc)*mc - 8*x)*y*Log[x/mc^2])/(32*E^(x/y)) + (mc^3*Pi^2*y^
2*Log[x/mc^2])/(4*E^(x/y))

________________________________________________________________________________________

Rubi [A]  time = 0.871562, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 107, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {12, 6742, 2199, 2194, 2177, 2178, 2176, 2554} \[ \frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )}{384 y}+\frac{1}{16} \pi ^2 (1-2 \text{mc}) \text{mc}^6 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{1}{32} \pi ^2 \text{mc}^3 y \left (-12 \text{mc}^2+3 \text{mc}-8 y\right ) \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 y^2 e^{-\frac{x}{y}}+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )+\frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 e^{-\frac{x}{y}}}{384 x}+\frac{3}{8} \pi ^2 \text{mc}^5 y e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 x y e^{-\frac{x}{y}}-\frac{1}{32} \pi ^2 \text{mc}^3 y (3 (1-4 \text{mc}) \text{mc}-8 x) e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )-\frac{1}{128} \pi ^2 (4 \text{mc}+1) x^2 y e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) y^3 e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) x y^2 e^{-\frac{x}{y}} \]

Antiderivative was successfully verified.

[In]

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*m
c*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]

[Out]

((3 - 4*mc)*mc^8*Pi^2)/(384*E^(x/y)*x) + (3*mc^5*Pi^2*y)/(8*E^(x/y)) + ((3 - 22*mc)*mc^2*Pi^2*x*y)/(48*E^(x/y)
) - ((1 + 4*mc)*Pi^2*x^2*y)/(128*E^(x/y)) + ((3 - 22*mc)*mc^2*Pi^2*y^2)/(48*E^(x/y)) + (mc^3*Pi^2*y^2)/(4*E^(x
/y)) - ((1 + 4*mc)*Pi^2*x*y^2)/(64*E^(x/y)) - ((1 + 4*mc)*Pi^2*y^3)/(64*E^(x/y)) + ((1 - 2*mc)*mc^6*Pi^2*ExpIn
tegralEi[-(x/y)])/16 + ((3 - 4*mc)*mc^8*Pi^2*ExpIntegralEi[-(x/y)])/(384*y) + (mc^3*Pi^2*(3*mc - 12*mc^2 - 8*y
)*y*ExpIntegralEi[-(x/y)])/32 - (mc^3*Pi^2*(3*(1 - 4*mc)*mc - 8*x)*y*Log[x/mc^2])/(32*E^(x/y)) + (mc^3*Pi^2*y^
2*Log[x/mc^2])/(4*E^(x/y))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rubi steps

\begin{align*} \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} \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 )}{x^2} \, dx\\ &=\frac{1}{384} \int \left (\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}) \text{mc}^4 x+(21-188 \text{mc}) \text{mc}^2 x^2-3 (1+4 \text{mc}) x^3\right )}{x^2}-12 e^{-\frac{x}{y}} \text{mc}^3 \pi ^2 \left (-3 \text{mc}+12 \text{mc}^2+8 x\right ) \log \left (\frac{x}{\text{mc}^2}\right )\right ) \, dx\\ &=\frac{1}{384} \pi ^2 \int \frac{e^{-\frac{x}{y}} \left (\text{mc}^2-x\right ) \left (-(3-4 \text{mc}) \text{mc}^6+(21-44 \text{mc}) \text{mc}^4 x+(21-188 \text{mc}) \text{mc}^2 x^2-3 (1+4 \text{mc}) x^3\right )}{x^2} \, dx-\frac{1}{32} \left (\text{mc}^3 \pi ^2\right ) \int e^{-\frac{x}{y}} \left (-3 \text{mc}+12 \text{mc}^2+8 x\right ) \log \left (\frac{x}{\text{mc}^2}\right ) \, dx\\ &=-\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 )+\frac{1}{384} \pi ^2 \int \left (-144 e^{-\frac{x}{y}} \text{mc}^5+\frac{e^{-\frac{x}{y}} \text{mc}^8 (-3+4 \text{mc})}{x^2}-\frac{24 e^{-\frac{x}{y}} \text{mc}^6 (-1+2 \text{mc})}{x}+8 e^{-\frac{x}{y}} \text{mc}^2 (-3+22 \text{mc}) x+3 e^{-\frac{x}{y}} (1+4 \text{mc}) x^2\right ) \, dx+\frac{1}{32} \left (\text{mc}^3 \pi ^2\right ) \int \frac{e^{-\frac{x}{y}} \left (3 \text{mc}-12 \text{mc}^2-8 x-8 y\right ) y}{x} \, dx\\ &=-\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 )-\frac{1}{48} \left ((3-22 \text{mc}) \text{mc}^2 \pi ^2\right ) \int e^{-\frac{x}{y}} x \, dx-\frac{1}{8} \left (3 \text{mc}^5 \pi ^2\right ) \int e^{-\frac{x}{y}} \, dx+\frac{1}{16} \left ((1-2 \text{mc}) \text{mc}^6 \pi ^2\right ) \int \frac{e^{-\frac{x}{y}}}{x} \, dx-\frac{1}{384} \left ((3-4 \text{mc}) \text{mc}^8 \pi ^2\right ) \int \frac{e^{-\frac{x}{y}}}{x^2} \, dx+\frac{1}{128} \left ((1+4 \text{mc}) \pi ^2\right ) \int e^{-\frac{x}{y}} x^2 \, dx+\frac{1}{32} \left (\text{mc}^3 \pi ^2 y\right ) \int \frac{e^{-\frac{x}{y}} \left (3 \text{mc}-12 \text{mc}^2-8 x-8 y\right )}{x} \, 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}{16} (1-2 \text{mc}) \text{mc}^6 \pi ^2 \text{Ei}\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 )+\frac{\left ((3-4 \text{mc}) \text{mc}^8 \pi ^2\right ) \int \frac{e^{-\frac{x}{y}}}{x} \, dx}{384 y}-\frac{1}{48} \left ((3-22 \text{mc}) \text{mc}^2 \pi ^2 y\right ) \int e^{-\frac{x}{y}} \, dx+\frac{1}{32} \left (\text{mc}^3 \pi ^2 y\right ) \int \left (-8 e^{-\frac{x}{y}}+\frac{e^{-\frac{x}{y}} \left (3 \text{mc}-12 \text{mc}^2-8 y\right )}{x}\right ) \, dx+\frac{1}{64} \left ((1+4 \text{mc}) \pi ^2 y\right ) \int e^{-\frac{x}{y}} x \, 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}{64} e^{-\frac{x}{y}} (1+4 \text{mc}) \pi ^2 x y^2+\frac{1}{16} (1-2 \text{mc}) \text{mc}^6 \pi ^2 \text{Ei}\left (-\frac{x}{y}\right )+\frac{(3-4 \text{mc}) \text{mc}^8 \pi ^2 \text{Ei}\left (-\frac{x}{y}\right )}{384 y}-\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 )-\frac{1}{4} \left (\text{mc}^3 \pi ^2 y\right ) \int e^{-\frac{x}{y}} \, dx+\frac{1}{32} \left (\text{mc}^3 \pi ^2 \left (3 \text{mc}-12 \text{mc}^2-8 y\right ) y\right ) \int \frac{e^{-\frac{x}{y}}}{x} \, dx+\frac{1}{64} \left ((1+4 \text{mc}) \pi ^2 y^2\right ) \int e^{-\frac{x}{y}} \, 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 \text{Ei}\left (-\frac{x}{y}\right )+\frac{(3-4 \text{mc}) \text{mc}^8 \pi ^2 \text{Ei}\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 \text{Ei}\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 )\\ \end{align*}

Mathematica [A]  time = 0.139802, size = 181, normalized size = 0.55 \[ \frac{1}{384} \pi ^2 \left (\frac{e^{-\frac{x}{y}} \left (144 \text{mc}^5 x y-16 \text{mc}^3 x y (11 x+5 y)+24 \text{mc}^2 x y (x+y)+12 \text{mc}^3 x y \left (12 \text{mc}^2-3 \text{mc}+8 (x+y)\right ) \log \left (\frac{x}{\text{mc}^2}\right )-4 \text{mc}^9+3 \text{mc}^8-12 \text{mc} x y \left (x^2+2 x y+2 y^2\right )-3 x y \left (x^2+2 x y+2 y^2\right )\right )}{x}-\frac{\text{mc}^3 \left (144 \text{mc}^2 y^2+48 \text{mc}^4 y-24 \text{mc}^3 y+4 \text{mc}^6-3 \text{mc}^5-36 \text{mc} y^2+96 y^3\right ) \text{ExpIntegralEi}\left (-\frac{x}{y}\right )}{y}\right ) \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(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

________________________________________________________________________________________

Maple [C]  time = 0.144, size = 1356, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*m
c^3*Pi^2*(-12*mc^2+3*mc-8*x)*x^2*ln(x/mc^2))/exp(x/y)/x^2,x)

[Out]

1/4*Pi^2*mc^3*y^2*Ei(1,x/y)-3/32*Pi^2*mc^4*y*Ei(1,x/y)+3/8*Pi^2*mc^5*y*Ei(1,x/y)-3/64*I*y*Pi^3*exp(-x/y)*mc^4*
csgn(I*x)*csgn(I/mc^2*x)^2+1/8*I*y^2*Pi^3*mc^3*csgn(I*x)*csgn(I/mc^2*x)^2*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)-1/8*I*y*Pi^3*mc^3*csgn(I/mc^2*x)^3*x*exp(-x/y)+1/8*I*y*Pi^3*mc^3*csgn(I*mc^2
)^3*x*exp(-x/y)-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)+3/16*I*y*Pi^3*exp(-x/y)*mc^5*csgn(I/mc^2)*csgn(I/mc^2*x)^2+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*x)*csgn(I/mc^2*x)^2-3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2)*csgn(I/mc^2*x)^2-1/16*Pi^2*mc^6*Ei(1,x/y)+1/8*
Pi^2*mc^7*Ei(1,x/y)-1/64*y^3*Pi^2*exp(-x/y)+1/384*(144*Pi^2*mc^5*y-36*Pi^2*mc^4*y+96*Pi^2*mc^3*x*y+96*Pi^2*mc^
3*y^2)*exp(-x/y)*ln(x)-1/8*I*y*Pi^3*mc^3*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)*x*exp(-x/y)-1/2*y*Pi^2*ln(mc)*m
c^3*x*exp(-x/y)-1/8*I*y^2*Pi^3*mc^3*csgn(I/mc^2*x)^3*exp(-x/y)+1/8*I*y^2*Pi^3*mc^3*csgn(I*mc^2)^3*exp(-x/y)-3/
64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I*mc^2)^3+3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2*x)^3+3/16*I*y*Pi^3*exp(-x/y
)*mc^5*csgn(I*mc^2)^3-3/16*I*y*Pi^3*exp(-x/y)*mc^5*csgn(I/mc^2*x)^3-3/64*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*csgn(I*mc)*csgn(I*mc^2)^2-1/128*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^5-1/128/y*Pi^2*mc^8
*Ei(1,x/y)+1/96/y*Pi^2*mc^9*Ei(1,x/y)-1/96*Pi^2*mc^9/x*exp(-x/y)+1/128*Pi^2*mc^8/x*exp(-x/y)-5/24*mc^3*Pi^2*y^
2*exp(-x/y)+1/8*I*y*Pi^3*mc^3*csgn(I*mc)^2*csgn(I*mc^2)*x*exp(-x/y)-1/4*I*y*Pi^3*mc^3*csgn(I*mc)*csgn(I*mc^2)^
2*x*exp(-x/y)+1/8*I*y*Pi^3*mc^3*csgn(I*x)*csgn(I/mc^2*x)^2*x*exp(-x/y)+1/8*I*y*Pi^3*mc^3*csgn(I/mc^2)*csgn(I/m
c^2*x)^2*x*exp(-x/y)-1/8*I*y^2*Pi^3*mc^3*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)*exp(-x/y)-3/16*I*y*Pi^3*exp(-x/
y)*mc^5*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)+3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*
x)+1/16*y*Pi^2*mc^2*x*exp(-x/y)-11/24*y*Pi^2*mc^3*x*exp(-x/y)-1/2*y^2*Pi^2*ln(mc)*mc^3*exp(-x/y)-1/32*y*Pi^2*m
c*exp(-x/y)*x^2-1/16*y^2*Pi^2*mc*x*exp(-x/y)-3/4*y*Pi^2*exp(-x/y)*ln(mc)*mc^5+3/16*y*Pi^2*exp(-x/y)*ln(mc)*mc^
4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\pi ^{2} \mathit{mc}^{9} \Gamma \left (-1, \frac{x}{y}\right )}{96 \, y} - \frac{1}{8} \, \pi ^{2} \mathit{mc}^{7}{\rm Ei}\left (-\frac{x}{y}\right ) + \frac{\pi ^{2} \mathit{mc}^{8} \Gamma \left (-1, \frac{x}{y}\right )}{128 \, y} + \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) + \frac{1}{16} \, \pi ^{2} \mathit{mc}^{6}{\rm Ei}\left (-\frac{x}{y}\right ) - \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y{\rm Ei}\left (-\frac{x}{y}\right ) + \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y e^{\left (-\frac{x}{y}\right )} - \frac{3}{32} \, \pi ^{2} \mathit{mc}^{4} y e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) + \frac{3}{32} \, \pi ^{2} \mathit{mc}^{4} y{\rm Ei}\left (-\frac{x}{y}\right ) - \frac{11}{24} \, \pi ^{2}{\left (x y + y^{2}\right )} \mathit{mc}^{3} e^{\left (-\frac{x}{y}\right )} + \frac{1}{4} \, \pi ^{2}{\left ({\left (x y + y^{2}\right )} e^{\left (-\frac{x}{y}\right )} \log \left (x\right ) + \int \frac{{\left (2 \, x^{2} \log \left (\mathit{mc}\right ) - x y - y^{2}\right )} e^{\left (-\frac{x}{y}\right )}}{x}\,{d x}\right )} \mathit{mc}^{3} + \frac{1}{16} \, \pi ^{2}{\left (x y + y^{2}\right )} \mathit{mc}^{2} e^{\left (-\frac{x}{y}\right )} - \frac{1}{32} \, \pi ^{2}{\left (x^{2} y + 2 \, x y^{2} + 2 \, y^{3}\right )} \mathit{mc} e^{\left (-\frac{x}{y}\right )} - \frac{1}{128} \, \pi ^{2}{\left (x^{2} y + 2 \, x y^{2} + 2 \, y^{3}\right )} e^{\left (-\frac{x}{y}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

-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*p
i^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)

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Fricas [A]  time = 1.98407, size = 590, normalized size = 1.79 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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Sympy [A]  time = 16.2833, size = 330, normalized size = 1. \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*m
c*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)

[Out]

-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
*mc**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*exp(-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 + p
i**2*(-x**2*y*exp(-x/y) - 2*x*y**2*exp(-x/y) - 2*y**3*exp(-x/y))/128

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Giac [A]  time = 1.14092, size = 637, normalized size = 1.93 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

-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)