∫ −2ex+2e−x(−2x+2x2) x−50exx3+ee−x(−2x+2x2) (−4x+12x2−4x3+ex(−2+20x2))__________________________________________________________ ex+2e−x(−2x+2x2)x4+ex+e−x(−2x+2x2)(4x3−10x5)+ex(4x2−20x4+25x6) dx

3.102.12 $\int \frac {-2 e^{x+2 e^{-x} (-2 x+2 x^2)} x-50 e^x x^3+e^{e^{-x} (-2 x+2 x^2)} (-4 x+12 x^2-4 x^3+e^x (-2+20 x^2))}{e^{x+2 e^{-x} (-2 x+2 x^2)} x^4+e^{x+e^{-x} (-2 x+2 x^2)} (4 x^3-10 x^5)+e^x (4 x^2-20 x^4+25 x^6)} \, dx$

Optimal. Leaf size=30 \[ \frac {1}{\frac {2}{-5+\frac {e^{e^{-x} x (-2+2 x)}}{x}}+x^2} \]

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Rubi [F] time = 28.35, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2*x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 + E^x*(-

2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4*x^2 -

20*x^4 + 25*x^6)),x]

[Out]

x^(-2) - 4*Defer[Int][1/(E^x*(2 + E^((2*(-1 + x)*x)/E^x)*x - 5*x^2)), x] - 6*Defer[Int][1/(x^3*(-2 - E^((2*(-1

+ x)*x)/E^x)*x + 5*x^2)), x] + 4*Defer[Int][1/(E^x*x^2*(-2 - E^((2*(-1 + x)*x)/E^x)*x + 5*x^2)), x] - 12*Defe

r[Int][1/(E^x*x*(-2 - E^((2*(-1 + x)*x)/E^x)*x + 5*x^2)), x] - 12*Defer[Int][1/(E^(((-4 + E^x)*x)/E^x)*(-2*E^(

(2*x)/E^x) - E^((2*x^2)/E^x)*x + 5*E^((2*x)/E^x)*x^2)^2), x] + 8*Defer[Int][1/(E^(((-4 + E^x)*x)/E^x)*x^2*(-2*

E^((2*x)/E^x) - E^((2*x^2)/E^x)*x + 5*E^((2*x)/E^x)*x^2)^2), x] - 24*Defer[Int][1/(E^(((-4 + E^x)*x)/E^x)*x*(-

2*E^((2*x)/E^x) - E^((2*x^2)/E^x)*x + 5*E^((2*x)/E^x)*x^2)^2), x] + 60*Defer[Int][x/(E^(((-4 + E^x)*x)/E^x)*(-

2*E^((2*x)/E^x) - E^((2*x^2)/E^x)*x + 5*E^((2*x)/E^x)*x^2)^2), x] - 20*Defer[Int][x^2/(E^(((-4 + E^x)*x)/E^x)*

(-2*E^((2*x)/E^x) - E^((2*x^2)/E^x)*x + 5*E^((2*x)/E^x)*x^2)^2), x] - 4*Defer[Int][E^((4*x)/E^x)/(x^3*(E^((2*x

^2)/E^x)*x + E^((2*x)/E^x)*(2 - 5*x^2))^2), x] - 10*Defer[Int][E^((4*x)/E^x)/(x*(E^((2*x^2)/E^x)*x + E^((2*x)/

E^x)*(2 - 5*x^2))^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} \left (-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )\right )}{x^2 \left (2 e^{2 e^{-x} x}+e^{2 e^{-x} x^2} x-5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\int \left (-\frac {2}{x^3}+\frac {2 e^{-2 e^{-x} x-e^{-x} \left (-4+e^x\right ) x} \left (-3 e^x+2 x-6 x^2+2 x^3\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )}-\frac {2 e^{-e^{-x} \left (-4+e^x\right ) x} \left (2 e^x-4 x+12 x^2+5 e^x x^2+6 x^3-30 x^4+10 x^5\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}\right ) \, dx\\ &=\frac {1}{x^2}+2 \int \frac {e^{-2 e^{-x} x-e^{-x} \left (-4+e^x\right ) x} \left (-3 e^x+2 x-6 x^2+2 x^3\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )} \, dx-2 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} \left (2 e^x-4 x+12 x^2+5 e^x x^2+6 x^3-30 x^4+10 x^5\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\frac {1}{x^2}+2 \int \frac {e^{-x} \left (3 e^x-2 x \left (1-3 x+x^2\right )\right )}{x^3 \left (2+e^{2 e^{-x} (-1+x) x} x-5 x^2\right )} \, dx-2 \int \left (\frac {6 e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {2 e^{x-e^{-x} \left (-4+e^x\right ) x}}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}-\frac {4 e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {12 e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {5 e^{x-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}-\frac {30 e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {10 e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}\right ) \, dx\\ &=\frac {1}{x^2}+2 \int \left (-\frac {2 e^{-x}}{2+e^{2 e^{-x} (-1+x) x} x-5 x^2}-\frac {3}{x^3 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}+\frac {2 e^{-x}}{x^2 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}-\frac {6 e^{-x}}{x \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}\right ) \, dx-4 \int \frac {e^{x-e^{-x} \left (-4+e^x\right ) x}}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+8 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-10 \int \frac {e^{x-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-12 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-20 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-24 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+60 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\frac {1}{x^2}-4 \int \frac {e^{-x}}{2+e^{2 e^{-x} (-1+x) x} x-5 x^2} \, dx+4 \int \frac {e^{-x}}{x^2 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx-4 \int \frac {e^{4 e^{-x} x}}{x^3 \left (e^{2 e^{-x} x^2} x+e^{2 e^{-x} x} \left (2-5 x^2\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx+8 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-10 \int \frac {e^{4 e^{-x} x}}{x \left (e^{2 e^{-x} x^2} x+e^{2 e^{-x} x} \left (2-5 x^2\right )\right )^2} \, dx-12 \int \frac {e^{-x}}{x \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx-12 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-20 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-24 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+60 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Integrate[(-2*E^(x + (2*(-2*x + 2*x^2))/E^x)*x - 50*E^x*x^3 + E^((-2*x + 2*x^2)/E^x)*(-4*x + 12*x^2 - 4*x^3 +

E^x*(-2 + 20*x^2)))/(E^(x + (2*(-2*x + 2*x^2))/E^x)*x^4 + E^(x + (-2*x + 2*x^2)/E^x)*(4*x^3 - 10*x^5) + E^x*(4

*x^2 - 20*x^4 + 25*x^6)),x]

[Out]

$Aborted

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fricas [B] time = 0.51, size = 68, normalized size = 2.27 \begin {gather*} -\frac {5 \, x e^{x} - e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )}}{x^{2} e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-

50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20

*x^4+4*x^2)*exp(x)),x, algorithm="fricas")

[Out]

-(5*x*e^x - e^((2*x^2 + x*e^x - 2*x)*e^(-x)))/(x^2*e^((2*x^2 + x*e^x - 2*x)*e^(-x)) - (5*x^3 - 2*x)*e^x)

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giac [B] time = 0.54, size = 4292, normalized size = 143.07 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-

50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20

*x^4+4*x^2)*exp(x)),x, algorithm="giac")

[Out]

(12500*x^15*e^(4*x*e^(-x) + 3/2*x) - 7500*x^14*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 75000*x^14*e^(4*x*e^(-x

) + 3/2*x) + 1500*x^13*e^(4*x^2*e^(-x) + 3/2*x) + 45000*x^13*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 117500*x^

13*e^(4*x*e^(-x) + 3/2*x) - 100*x^12*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 9000*x^12*e^(4*x^2*e^(-x) + 3/2*x

) - 72500*x^12*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 12500*x^12*e^(4*x*e^(-x) + 5/2*x) + 45000*x^12*e^(4*x*e

^(-x) + 3/2*x) + 600*x^11*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 14900*x^11*e^(4*x^2*e^(-x) + 3/2*x) - 7500*x

^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 15000*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 37500*x^11*e^(4

*x*e^(-x) + 5/2*x) - 195500*x^11*e^(4*x*e^(-x) + 3/2*x) - 1020*x^10*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 15

00*x^10*e^(4*x^2*e^(-x) + 5/2*x) + 600*x^10*e^(4*x^2*e^(-x) + 3/2*x) + 22500*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x)

+ 5/2*x) + 97700*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 2500*x^10*e^(4*x*e^(-x) + 5/2*x) + 48000*x^10*e

^(4*x*e^(-x) + 3/2*x) - 100*x^9*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) + 120*x^9*e^(6*x^2*e^(-x) - 2*x*e^(-x) +

3/2*x) - 4500*x^9*e^(4*x^2*e^(-x) + 5/2*x) - 15540*x^9*e^(4*x^2*e^(-x) + 3/2*x) - 3500*x^9*e^(2*x^2*e^(-x) +

2*x*e^(-x) + 5/2*x) - 31200*x^9*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 3125*x^9*e^(4*x*e^(-x) + 7/2*x) + 3000

0*x^9*e^(4*x*e^(-x) + 5/2*x) + 108800*x^9*e^(4*x*e^(-x) + 3/2*x) + 300*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*

x) + 764*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 1100*x^8*e^(4*x^2*e^(-x) + 5/2*x) + 6240*x^8*e^(4*x^2*e^(

-x) + 3/2*x) - 1875*x^8*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x) - 12000*x^8*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x

) - 41840*x^8*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 10000*x^8*e^(4*x*e^(-x) + 5/2*x) - 52800*x^8*e^(4*x*e^(-

x) + 3/2*x) - 100*x^7*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 384*x^7*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) +

375*x^7*e^(4*x^2*e^(-x) + 7/2*x) + 1200*x^7*e^(4*x^2*e^(-x) + 5/2*x) + 4496*x^7*e^(4*x^2*e^(-x) + 3/2*x) + 480

0*x^7*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 23040*x^7*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 22880*x^7*e^(4

*x*e^(-x) + 3/2*x) - 25*x^6*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 7/2*x) - 96*x^6*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*

x) - 640*x^6*e^(4*x^2*e^(-x) + 5/2*x) - 2976*x^6*e^(4*x^2*e^(-x) + 3/2*x) - 500*x^6*e^(2*x^2*e^(-x) + 2*x*e^(-

x) + 7/2*x) - 2400*x^6*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 5696*x^6*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x)

+ 1600*x^6*e^(4*x*e^(-x) + 5/2*x) + 17280*x^6*e^(4*x*e^(-x) + 3/2*x) + 16*x^5*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5

/2*x) + 96*x^5*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 200*x^5*e^(4*x^2*e^(-x) + 7/2*x) + 720*x^5*e^(4*x^2*e^(

-x) + 5/2*x) - 144*x^5*e^(4*x^2*e^(-x) + 3/2*x) + 160*x^5*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 5376*x^5*e^(

2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 1000*x^5*e^(4*x*e^(-x) + 7/2*x) - 4800*x^5*e^(4*x*e^(-x) + 5/2*x) + 320*x

^5*e^(4*x*e^(-x) + 3/2*x) - 20*x^4*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 7/2*x) - 48*x^4*e^(6*x^2*e^(-x) - 2*x*e^(-x)

+ 5/2*x) - 16*x^4*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 176*x^4*e^(4*x^2*e^(-x) + 5/2*x) + 384*x^4*e^(4*x^2

*e^(-x) + 3/2*x) + 400*x^4*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x) + 1920*x^4*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2

*x) + 256*x^4*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 1280*x^4*e^(4*x*e^(-x) + 5/2*x) - 1920*x^4*e^(4*x*e^(-x)

+ 3/2*x) + 16*x^3*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 20*x^3*e^(4*x^2*e^(-x) + 7/2*x) - 192*x^3*e^(4*x^2*

e^(-x) + 5/2*x) - 64*x^3*e^(4*x^2*e^(-x) + 3/2*x) - 576*x^3*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 384*x^3*e^

(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 960*x^3*e^(4*x*e^(-x) + 5/2*x) + 320*x^3*e^(4*x*e^(-x) + 3/2*x) - 4*x^2*

e^(6*x^2*e^(-x) - 2*x*e^(-x) + 7/2*x) + 64*x^2*e^(4*x^2*e^(-x) + 5/2*x) + 80*x^2*e^(2*x^2*e^(-x) + 2*x*e^(-x)

+ 7/2*x) - 192*x^2*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 64*x^2*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 320*

x^2*e^(4*x*e^(-x) + 5/2*x) - 16*x*e^(4*x^2*e^(-x) + 7/2*x) + 64*x*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 80*x

*e^(4*x*e^(-x) + 7/2*x) - 16*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x))/(12500*x^17*e^(4*x*e^(-x) + 3/2*x) - 7500*

x^16*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 75000*x^16*e^(4*x*e^(-x) + 3/2*x) + 1500*x^15*e^(4*x^2*e^(-x) + 3

/2*x) + 45000*x^15*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 112500*x^15*e^(4*x*e^(-x) + 3/2*x) - 100*x^14*e^(6*

x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 9000*x^14*e^(4*x^2*e^(-x) + 3/2*x) - 70500*x^14*e^(2*x^2*e^(-x) + 2*x*e^(-x

) + 3/2*x) + 12500*x^14*e^(4*x*e^(-x) + 5/2*x) + 75000*x^14*e^(4*x*e^(-x) + 3/2*x) + 600*x^13*e^(6*x^2*e^(-x)

- 2*x*e^(-x) + 3/2*x) + 14700*x^13*e^(4*x^2*e^(-x) + 3/2*x) - 7500*x^13*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x)

- 27000*x^13*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 37500*x^13*e^(4*x*e^(-x) + 5/2*x) - 242500*x^13*e^(4*x*e^

(-x) + 3/2*x) - 1020*x^12*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 1500*x^12*e^(4*x^2*e^(-x) + 5/2*x) + 1800*x^

12*e^(4*x^2*e^(-x) + 3/2*x) + 22500*x^12*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 117300*x^12*e^(2*x^2*e^(-x) +

2*x*e^(-x) + 3/2*x) - 2500*x^12*e^(4*x*e^(-x) + 5/2*x) + 30000*x^12*e^(4*x*e^(-x) + 3/2*x) - 100*x^11*e^(6*x^

2*e^(-x) - 2*x*e^(-x) + 5/2*x) + 120*x^11*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 4500*x^11*e^(4*x^2*e^(-x) +

5/2*x) - 17580*x^11*e^(4*x^2*e^(-x) + 3/2*x) - 1500*x^11*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 28800*x^11*e^

(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 3125*x^11*e^(4*x*e^(-x) + 7/2*x) + 45000*x^11*e^(4*x*e^(-x) + 5/2*x) + 1

87000*x^11*e^(4*x*e^(-x) + 3/2*x) + 300*x^10*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) + 764*x^10*e^(6*x^2*e^(-x)

- 2*x*e^(-x) + 3/2*x) + 900*x^10*e^(4*x^2*e^(-x) + 5/2*x) + 6480*x^10*e^(4*x^2*e^(-x) + 3/2*x) - 1875*x^10*e^(

2*x^2*e^(-x) + 2*x*e^(-x) + 7/2*x) - 18000*x^10*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 65280*x^10*e^(2*x^2*e^

(-x) + 2*x*e^(-x) + 3/2*x) - 11000*x^10*e^(4*x*e^(-x) + 5/2*x) - 72000*x^10*e^(4*x*e^(-x) + 3/2*x) - 100*x^9*e

^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 384*x^9*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) + 375*x^9*e^(4*x^2*e^(-x)

+ 7/2*x) + 1800*x^9*e^(4*x^2*e^(-x) + 5/2*x) + 6024*x^9*e^(4*x^2*e^(-x) + 3/2*x) + 6000*x^9*e^(2*x^2*e^(-x) +

2*x*e^(-x) + 5/2*x) + 31680*x^9*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 1250*x^9*e^(4*x*e^(-x) + 7/2*x) - 120

00*x^9*e^(4*x*e^(-x) + 5/2*x) - 66400*x^9*e^(4*x*e^(-x) + 3/2*x) - 25*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 7/2*x

) - 96*x^8*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*x) - 840*x^8*e^(4*x^2*e^(-x) + 5/2*x) - 3744*x^8*e^(4*x^2*e^(-x)

+ 3/2*x) + 13728*x^8*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 5600*x^8*e^(4*x*e^(-x) + 5/2*x) + 38400*x^8*e^(4

*x*e^(-x) + 3/2*x) + 16*x^7*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) + 96*x^7*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 3/2*

x) + 150*x^7*e^(4*x^2*e^(-x) + 7/2*x) + 720*x^7*e^(4*x^2*e^(-x) + 5/2*x) - 336*x^7*e^(4*x^2*e^(-x) + 3/2*x) -

960*x^7*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 10368*x^7*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 1000*x^7*e^(

4*x*e^(-x) + 7/2*x) - 4800*x^7*e^(4*x*e^(-x) + 5/2*x) + 9472*x^7*e^(4*x*e^(-x) + 3/2*x) - 20*x^6*e^(6*x^2*e^(-

x) - 2*x*e^(-x) + 7/2*x) - 48*x^6*e^(6*x^2*e^(-x) - 2*x*e^(-x) + 5/2*x) - 16*x^6*e^(6*x^2*e^(-x) - 2*x*e^(-x)

+ 3/2*x) - 144*x^6*e^(4*x^2*e^(-x) + 5/2*x) + 576*x^6*e^(4*x^2*e^(-x) + 3/2*x) + 600*x^6*e^(2*x^2*e^(-x) + 2*x

*e^(-x) + 7/2*x) + 2880*x^6*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 192*x^6*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2

*x) + 640*x^6*e^(4*x*e^(-x) + 5/2*x) - 8832*x^6*e^(4*x*e^(-x) + 3/2*x) + 16*x^5*e^(6*x^2*e^(-x) - 2*x*e^(-x) +

5/2*x) - 60*x^5*e^(4*x^2*e^(-x) + 7/2*x) - 288*x^5*e^(4*x^2*e^(-x) + 5/2*x) - 96*x^5*e^(4*x^2*e^(-x) + 3/2*x)

- 768*x^5*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 1152*x^5*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) + 400*x^5*e^

(4*x*e^(-x) + 7/2*x) + 2880*x^5*e^(4*x*e^(-x) + 5/2*x) + 192*x^5*e^(4*x*e^(-x) + 3/2*x) - 4*x^4*e^(6*x^2*e^(-x

) - 2*x*e^(-x) + 7/2*x) + 96*x^4*e^(4*x^2*e^(-x) + 5/2*x) - 576*x^4*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) - 19

2*x^4*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 3/2*x) - 832*x^4*e^(4*x*e^(-x) + 5/2*x) + 768*x^4*e^(4*x*e^(-x) + 3/2*x)

- 24*x^3*e^(4*x^2*e^(-x) + 7/2*x) + 192*x^3*e^(2*x^2*e^(-x) + 2*x*e^(-x) + 5/2*x) + 80*x^3*e^(4*x*e^(-x) + 7/2

*x) - 384*x^3*e^(4*x*e^(-x) + 5/2*x) - 128*x^3*e^(4*x*e^(-x) + 3/2*x) - 48*x^2*e^(2*x^2*e^(-x) + 2*x*e^(-x) +

7/2*x) + 128*x^2*e^(4*x*e^(-x) + 5/2*x) - 32*x*e^(4*x*e^(-x) + 7/2*x))

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maple [A] time = 0.07, size = 33, normalized size = 1.10


method	result	size

risch	$\frac {1}{x^{2}}+\frac {2}{x^{2} \left (5 x^{2}-x \,{\mathrm e}^{2 x \left (x -1\right ) {\mathrm e}^{-x}}-2\right )}$	$33$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-50*exp

(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20*x^4+4

*x^2)*exp(x)),x,method=_RETURNVERBOSE)

[Out]

1/x^2+2/x^2/(5*x^2-x*exp(2*x*(x-1)*exp(-x))-2)

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maxima [B] time = 0.42, size = 62, normalized size = 2.07 \begin {gather*} -\frac {5 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )}}{x^{2} e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)*exp((2*x^2-2*x)/exp(x))^2+((20*x^2-2)*exp(x)-4*x^3+12*x^2-4*x)*exp((2*x^2-2*x)/exp(x))-

50*exp(x)*x^3)/(x^4*exp(x)*exp((2*x^2-2*x)/exp(x))^2+(-10*x^5+4*x^3)*exp(x)*exp((2*x^2-2*x)/exp(x))+(25*x^6-20

*x^4+4*x^2)*exp(x)),x, algorithm="maxima")

[Out]

-(5*x*e^(2*x*e^(-x)) - e^(2*x^2*e^(-x)))/(x^2*e^(2*x^2*e^(-x)) - (5*x^3 - 2*x)*e^(2*x*e^(-x)))

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mupad [B] time = 8.75, size = 41, normalized size = 1.37 \begin {gather*} \frac {1}{x^2}-\frac {2}{2\,x^2-5\,x^4+x^3\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{-x}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(50*x^3*exp(x) + exp(-exp(-x)*(2*x - 2*x^2))*(4*x - exp(x)*(20*x^2 - 2) - 12*x^2 + 4*x^3) + 2*x*exp(-2*ex

p(-x)*(2*x - 2*x^2))*exp(x))/(exp(x)*(4*x^2 - 20*x^4 + 25*x^6) + exp(-exp(-x)*(2*x - 2*x^2))*exp(x)*(4*x^3 - 1

0*x^5) + x^4*exp(-2*exp(-x)*(2*x - 2*x^2))*exp(x)),x)

[Out]

1/x^2 - 2/(2*x^2 - 5*x^4 + x^3*exp(-2*x*exp(-x))*exp(2*x^2*exp(-x)))

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sympy [A] time = 0.31, size = 32, normalized size = 1.07 \begin {gather*} - \frac {2}{- 5 x^{4} + x^{3} e^{\left (2 x^{2} - 2 x\right ) e^{- x}} + 2 x^{2}} + \frac {1}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)*exp((2*x**2-2*x)/exp(x))**2+((20*x**2-2)*exp(x)-4*x**3+12*x**2-4*x)*exp((2*x**2-2*x)/ex

p(x))-50*exp(x)*x**3)/(x**4*exp(x)*exp((2*x**2-2*x)/exp(x))**2+(-10*x**5+4*x**3)*exp(x)*exp((2*x**2-2*x)/exp(x

))+(25*x**6-20*x**4+4*x**2)*exp(x)),x)

[Out]

-2/(-5*x**4 + x**3*exp((2*x**2 - 2*x)*exp(-x)) + 2*x**2) + x**(-2)

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3.102.12 \(\int \frac {-2 e^{x+2 e^{-x} (-2 x+2 x^2)} x-50 e^x x^3+e^{e^{-x} (-2 x+2 x^2)} (-4 x+12 x^2-4 x^3+e^x (-2+20 x^2))}{e^{x+2 e^{-x} (-2 x+2 x^2)} x^4+e^{x+e^{-x} (-2 x+2 x^2)} (4 x^3-10 x^5)+e^x (4 x^2-20 x^4+25 x^6)} \, dx\)