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

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

[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)*P
i^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*ExpIntegralEi[-(x/y)])/16 + ((3 - 4*mc)*mc^8*Pi^2*ExpIntegralEi[-(x/y)])/(3
84*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))

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Rubi [A]  time = 1.41925, 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 \[ \frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )}{384 y}+\frac{1}{16} \pi ^2 (1-2 \text{mc}) \text{mc}^6 \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{1}{32} \pi ^2 \text{mc}^3 y \left (-12 \text{mc}^2+3 \text{mc}-8 y\right ) \text{ExpIntegralEi}\left (-\frac{x}{y}\right )+\frac{\pi ^2 (3-4 \text{mc}) \text{mc}^8 e^{-\frac{x}{y}}}{384 x}+\frac{3}{8} \pi ^2 \text{mc}^5 y e^{-\frac{x}{y}}+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 y^2 e^{-\frac{x}{y}}+\frac{1}{48} \pi ^2 (3-22 \text{mc}) \text{mc}^2 x y e^{-\frac{x}{y}}+\frac{1}{4} \pi ^2 \text{mc}^3 y^2 e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )-\frac{1}{32} \pi ^2 \text{mc}^3 y (3 (1-4 \text{mc}) \text{mc}-8 x) e^{-\frac{x}{y}} \log \left (\frac{x}{\text{mc}^2}\right )-\frac{1}{128} \pi ^2 (4 \text{mc}+1) x^2 y e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) y^3 e^{-\frac{x}{y}}-\frac{1}{64} \pi ^2 (4 \text{mc}+1) x y^2 e^{-\frac{x}{y}} \]

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*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]

((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)*P
i^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*ExpIntegralEi[-(x/y)])/16 + ((3 - 4*mc)*mc^8*Pi^2*ExpIntegralEi[-(x/y)])/(3
84*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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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)

[Out]

Timed out

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Mathematica [A]  time = 0.219004, size = 181, normalized size = 0.55 \[ \frac{1}{384} \pi ^2 \left (\frac{e^{-\frac{x}{y}} \left (-4 \text{mc}^9+3 \text{mc}^8+144 \text{mc}^5 x y-16 \text{mc}^3 x y (11 x+5 y)+24 \text{mc}^2 x y (x+y)+12 \text{mc}^3 x y \left (12 \text{mc}^2-3 \text{mc}+8 (x+y)\right ) \log \left (\frac{x}{\text{mc}^2}\right )-12 \text{mc} x y \left (x^2+2 x y+2 y^2\right )-3 x y \left (x^2+2 x y+2 y^2\right )\right )}{x}-\frac{\text{mc}^3 \left (4 \text{mc}^6-3 \text{mc}^5+48 \text{mc}^4 y-24 \text{mc}^3 y+144 \text{mc}^2 y^2-36 \text{mc} y^2+96 y^3\right ) \text{ExpIntegralEi}\left (-\frac{x}{y}\right )}{y}\right ) \]

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*m
c^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

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Maple [C]  time = 0.086, size = 1356, normalized size = 4.1 \[ \text{result too large to display} \]

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

[Out]

-3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I*x)*csgn(I/mc^2*x)^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)^2-3/64*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+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+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^2*Pi^3*mc^3*csgn(I*x)*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)-1/4*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*Pi^3*mc^3*csgn(I*mc^2)^3*x*exp(-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^5-1/128/y*Pi^2*mc^8*Ei(1,x/y)+1/96/y*Pi^2*mc^9*Ei(1,x/y)
+1/128*Pi^2*mc^8/x*exp(-x/y)-1/96*Pi^2*mc^9/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*mc)^2*csgn(I*mc^2)*x*ex
p(-x/y)+3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)-3/16*
I*y*Pi^3*exp(-x/y)*mc^5*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)+1/8*I*y*Pi^3*mc^3*
csgn(I/mc^2)*csgn(I/mc^2*x)^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^2*Pi^3*mc^3*csgn(I/mc^2)*csgn(I*x)*csgn(I/mc^2*x)*exp
(-x/y)-5/24*mc^3*Pi^2*y^2*exp(-x/y)-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)-1/2*y*Pi^2*ln(mc)*mc^3*x*exp(-x/y)+3/16*I*y*Pi^3*ex
p(-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)^3+3/64*I*y*Pi^3*exp(-x/y)*mc^4*csgn(I/mc^2*x
)^3+1/8*I*y^2*Pi^3*mc^3*csgn(I*mc^2)^3*exp(-x/y)-1/8*I*y^2*Pi^3*mc^3*csgn(I/mc^2
*x)^3*exp(-x/y)+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)+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^2*Pi^2*ln(mc)*mc^3*exp(-x/y)-1/32*y*Pi^2*mc*exp(-x/y)*x^2-1/
16*y^2*Pi^2*mc*x*exp(-x/y)+1/16*y*Pi^2*mc^2*x*exp(-x/y)-11/24*y*Pi^2*mc^3*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 [A]  time = 1.59409, size = 435, normalized size = 1.32 \[ -\frac{\pi ^{2} \mathit{mc}^{9} \Gamma \left (-1, \frac{x}{y}\right )}{96 \, y} - \frac{1}{8} \, \pi ^{2} \mathit{mc}^{7}{\rm Ei}\left (-\frac{x}{y}\right ) + \frac{\pi ^{2} \mathit{mc}^{8} \Gamma \left (-1, \frac{x}{y}\right )}{128 \, y} + \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) + \frac{1}{16} \, \pi ^{2} \mathit{mc}^{6}{\rm Ei}\left (-\frac{x}{y}\right ) - \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y{\rm Ei}\left (-\frac{x}{y}\right ) + \frac{3}{8} \, \pi ^{2} \mathit{mc}^{5} y e^{\left (-\frac{x}{y}\right )} - \frac{3}{32} \, \pi ^{2} \mathit{mc}^{4} y e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) + \frac{3}{32} \, \pi ^{2} \mathit{mc}^{4} y{\rm Ei}\left (-\frac{x}{y}\right ) + \frac{1}{4} \, \pi ^{2}{\left (x y + y^{2}\right )} \mathit{mc}^{3} e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) - \frac{11}{24} \, \pi ^{2}{\left (x y + y^{2}\right )} \mathit{mc}^{3} e^{\left (-\frac{x}{y}\right )} - \frac{1}{4} \, \pi ^{2}{\left (y^{2}{\rm Ei}\left (-\frac{x}{y}\right ) - y^{2} e^{\left (-\frac{x}{y}\right )}\right )} \mathit{mc}^{3} + \frac{1}{16} \, \pi ^{2}{\left (x y + y^{2}\right )} \mathit{mc}^{2} e^{\left (-\frac{x}{y}\right )} - \frac{1}{32} \, \pi ^{2}{\left (x^{2} y + 2 \, x y^{2} + 2 \, y^{3}\right )} \mathit{mc} e^{\left (-\frac{x}{y}\right )} - \frac{1}{128} \, \pi ^{2}{\left (x^{2} y + 2 \, x y^{2} + 2 \, y^{3}\right )} e^{\left (-\frac{x}{y}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/384*(12*pi^2*(12*mc^2 - 3*mc + 8*x)*mc^3*x^2*log(x/mc^2) - 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))*e^(-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*gamm
a(-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) + 1/4*pi^2*(x*y + y^2)*mc^3*e^(-x/y)*log
(x/mc^2) - 11/24*pi^2*(x*y + y^2)*mc^3*e^(-x/y) - 1/4*pi^2*(y^2*Ei(-x/y) - y^2*e
^(-x/y))*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 = 0.225735, size = 363, normalized size = 1.1 \[ \frac{12 \,{\left (8 \, \pi ^{2} \mathit{mc}^{3} x y^{3} +{\left (8 \, \pi ^{2} \mathit{mc}^{3} x^{2} + 3 \, \pi ^{2}{\left (4 \, \mathit{mc}^{5} - \mathit{mc}^{4}\right )} x\right )} y^{2}\right )} e^{\left (-\frac{x}{y}\right )} \log \left (\frac{x}{\mathit{mc}^{2}}\right ) -{\left (96 \, \pi ^{2} \mathit{mc}^{3} x y^{3} + 36 \, \pi ^{2}{\left (4 \, \mathit{mc}^{5} - \mathit{mc}^{4}\right )} x y^{2} + 24 \, \pi ^{2}{\left (2 \, \mathit{mc}^{7} - \mathit{mc}^{6}\right )} x y + \pi ^{2}{\left (4 \, \mathit{mc}^{9} - 3 \, \mathit{mc}^{8}\right )} x\right )}{\rm Ei}\left (-\frac{x}{y}\right ) -{\left (6 \, \pi ^{2}{\left (4 \, \mathit{mc} + 1\right )} x y^{4} + \pi ^{2}{\left (4 \, \mathit{mc}^{9} - 3 \, \mathit{mc}^{8}\right )} y + 2 \,{\left (3 \, \pi ^{2}{\left (4 \, \mathit{mc} + 1\right )} x^{2} + 4 \, \pi ^{2}{\left (10 \, \mathit{mc}^{3} - 3 \, \mathit{mc}^{2}\right )} x\right )} y^{3} -{\left (144 \, \pi ^{2} \mathit{mc}^{5} x - 3 \, \pi ^{2}{\left (4 \, \mathit{mc} + 1\right )} x^{3} - 8 \, \pi ^{2}{\left (22 \, \mathit{mc}^{3} - 3 \, \mathit{mc}^{2}\right )} x^{2}\right )} y^{2}\right )} e^{\left (-\frac{x}{y}\right )}}{384 \, x y} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/384*(12*pi^2*(12*mc^2 - 3*mc + 8*x)*mc^3*x^2*log(x/mc^2) - 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))*e^(-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 [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\pi ^{2} \left (\int \left (- 144 mc^{5} e^{- \frac{x}{y}}\right )\, dx + \int 3 x^{2} e^{- \frac{x}{y}}\, dx + \int 12 mc x^{2} e^{- \frac{x}{y}}\, dx + \int \left (- 24 mc^{2} x e^{- \frac{x}{y}}\right )\, dx + \int 176 mc^{3} x e^{- \frac{x}{y}}\, dx + \int 36 mc^{4} e^{- \frac{x}{y}} \log{\left (\frac{x}{mc^{2}} \right )}\, dx + \int \left (- 144 mc^{5} e^{- \frac{x}{y}} \log{\left (\frac{x}{mc^{2}} \right )}\right )\, dx + \int \frac{24 mc^{6} e^{- \frac{x}{y}}}{x}\, dx + \int \left (- \frac{48 mc^{7} e^{- \frac{x}{y}}}{x}\right )\, dx + \int \left (- \frac{3 mc^{8} e^{- \frac{x}{y}}}{x^{2}}\right )\, dx + \int \frac{4 mc^{9} e^{- \frac{x}{y}}}{x^{2}}\, dx + \int \left (- 96 mc^{3} x e^{- \frac{x}{y}} \log{\left (\frac{x}{mc^{2}} \right )}\right )\, dx\right )}{384} \]

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*ln(x/mc**2))/exp(x/y)/x**2,x)

[Out]

pi**2*(Integral(-144*mc**5*exp(-x/y), x) + Integral(3*x**2*exp(-x/y), x) + Integ
ral(12*mc*x**2*exp(-x/y), x) + Integral(-24*mc**2*x*exp(-x/y), x) + Integral(176
*mc**3*x*exp(-x/y), x) + Integral(36*mc**4*exp(-x/y)*log(x/mc**2), x) + Integral
(-144*mc**5*exp(-x/y)*log(x/mc**2), x) + Integral(24*mc**6*exp(-x/y)/x, x) + Int
egral(-48*mc**7*exp(-x/y)/x, x) + Integral(-3*mc**8*exp(-x/y)/x**2, x) + Integra
l(4*mc**9*exp(-x/y)/x**2, x) + Integral(-96*mc**3*x*exp(-x/y)*log(x/mc**2), x))/
384

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GIAC/XCAS [A]  time = 0.215655, size = 637, normalized size = 1.93 \[ -\frac{4 \, \pi ^{2} \mathit{mc}^{9} x{\rm Ei}\left (-\frac{x}{y}\right ) + 4 \, \pi ^{2} \mathit{mc}^{9} y e^{\left (-\frac{x}{y}\right )} - 3 \, \pi ^{2} \mathit{mc}^{8} x{\rm Ei}\left (-\frac{x}{y}\right ) + 48 \, \pi ^{2} \mathit{mc}^{7} x y{\rm Ei}\left (-\frac{x}{y}\right ) - 3 \, \pi ^{2} \mathit{mc}^{8} y e^{\left (-\frac{x}{y}\right )} - 144 \, \pi ^{2} \mathit{mc}^{5} x y^{2} e^{\left (-\frac{x}{y}\right )}{\rm ln}\left (\frac{x}{\mathit{mc}^{2}}\right ) - 24 \, \pi ^{2} \mathit{mc}^{6} x y{\rm Ei}\left (-\frac{x}{y}\right ) + 144 \, \pi ^{2} \mathit{mc}^{5} x y^{2}{\rm Ei}\left (-\frac{x}{y}\right ) - 144 \, \pi ^{2} \mathit{mc}^{5} x y^{2} e^{\left (-\frac{x}{y}\right )} + 36 \, \pi ^{2} \mathit{mc}^{4} x y^{2} e^{\left (-\frac{x}{y}\right )}{\rm ln}\left (\frac{x}{\mathit{mc}^{2}}\right ) - 96 \, \pi ^{2} \mathit{mc}^{3} x^{2} y^{2} e^{\left (-\frac{x}{y}\right )}{\rm ln}\left (\frac{x}{\mathit{mc}^{2}}\right ) - 96 \, \pi ^{2} \mathit{mc}^{3} x y^{3} e^{\left (-\frac{x}{y}\right )}{\rm ln}\left (\frac{x}{\mathit{mc}^{2}}\right ) - 36 \, \pi ^{2} \mathit{mc}^{4} x y^{2}{\rm Ei}\left (-\frac{x}{y}\right ) + 96 \, \pi ^{2} \mathit{mc}^{3} x y^{3}{\rm Ei}\left (-\frac{x}{y}\right ) + 176 \, \pi ^{2} \mathit{mc}^{3} x^{2} y^{2} e^{\left (-\frac{x}{y}\right )} + 80 \, \pi ^{2} \mathit{mc}^{3} x y^{3} e^{\left (-\frac{x}{y}\right )} - 24 \, \pi ^{2} \mathit{mc}^{2} x^{2} y^{2} e^{\left (-\frac{x}{y}\right )} + 12 \, \pi ^{2} \mathit{mc} x^{3} y^{2} e^{\left (-\frac{x}{y}\right )} - 24 \, \pi ^{2} \mathit{mc}^{2} x y^{3} e^{\left (-\frac{x}{y}\right )} + 24 \, \pi ^{2} \mathit{mc} x^{2} y^{3} e^{\left (-\frac{x}{y}\right )} + 24 \, \pi ^{2} \mathit{mc} x y^{4} e^{\left (-\frac{x}{y}\right )} + 3 \, \pi ^{2} x^{3} y^{2} e^{\left (-\frac{x}{y}\right )} + 6 \, \pi ^{2} x^{2} y^{3} e^{\left (-\frac{x}{y}\right )} + 6 \, \pi ^{2} x y^{4} e^{\left (-\frac{x}{y}\right )}}{384 \, x y} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/384*(12*pi^2*(12*mc^2 - 3*mc + 8*x)*mc^3*x^2*log(x/mc^2) - 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))*e^(-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)*ln(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)*ln(x/mc^2) - 96*pi^2*mc^
3*x^2*y^2*e^(-x/y)*ln(x/mc^2) - 96*pi^2*mc^3*x*y^3*e^(-x/y)*ln(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)