3.23.18 \(\int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} (e^{4 x} (-6 x-12 x^2)+e^{8 e^{4-e^{x^2}}+8 x} (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3)+e^{4 e^{4-e^{x^2}}+4 x} (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} (-36 x-108 x^2))+e^{2 e^{4-e^{x^2}}+2 x} (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} (24 x+60 x^2))+e^{6 e^{4-e^{x^2}}+6 x} (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x (24 x+84 x^2))) \, dx\)

Optimal. Leaf size=34 \[ 3 e^{-\left (e^x-e^{2 \left (e^{4-e^{x^2}}+x\right )}\right )^4 x^2} \]

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Rubi [F]  time = 47.45, 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 \exp \left (-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2\right ) \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^
2) + 7*x)*x^2 - E^(8*E^(4 - E^x^2) + 8*x)*x^2)*(E^(4*x)*(-6*x - 12*x^2) + E^(8*E^(4 - E^x^2) + 8*x)*(-6*x - 24
*x^2 + 48*E^(4 - E^x^2 + x^2)*x^3) + E^(4*E^(4 - E^x^2) + 4*x)*(144*E^(4 - E^x^2 + 2*x + x^2)*x^3 + E^(2*x)*(-
36*x - 108*x^2)) + E^(2*E^(4 - E^x^2) + 2*x)*(-48*E^(4 - E^x^2 + 3*x + x^2)*x^3 + E^(3*x)*(24*x + 60*x^2)) + E
^(6*E^(4 - E^x^2) + 6*x)*(-144*E^(4 - E^x^2 + x + x^2)*x^3 + E^x*(24*x + 84*x^2))),x]

[Out]

-6*Defer[Int][E^(4*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*
E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x, x] + 24*Defer[Int][E^(2*E^(4 - E^x^2) + 5*x - E
^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2
+ 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x, x] - 36*Defer[Int][E^(4*E^(4 - E^x^2) + 6*x - E^(4*x)*x^2 - E^(8*(E^(4
- E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2)
+ 7*x)*x^2)*x, x] + 24*Defer[Int][E^(6*E^(4 - E^x^2) + 7*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E
^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x, x] - 6*De
fer[Int][E^(8*E^(4 - E^x^2) + 8*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*
x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x, x] - 12*Defer[Int][E^(4*x - E^(4*x
)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*
E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^2, x] + 60*Defer[Int][E^(2*E^(4 - E^x^2) + 5*x - E^(4*x)*x^2 - E^(8*(E^(4 - E
^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7
*x)*x^2)*x^2, x] - 108*Defer[Int][E^(4*E^(4 - E^x^2) + 6*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E
^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^2, x] + 84
*Defer[Int][E^(6*E^(4 - E^x^2) + 7*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*
x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^2, x] - 24*Defer[Int][E^(8*E^(4
- E^x^2) + 8*x - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 -
E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^2, x] - 48*Defer[Int][E^(4 + 2*E^(4 - E^x^2) - E^x^2 +
5*x + x^2 - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x
^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^3, x] + 144*Defer[Int][E^(4 + 4*E^(4 - E^x^2) - E^x^2 + 6*
x + x^2 - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2)
+ 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^3, x] - 144*Defer[Int][E^(4 + 6*E^(4 - E^x^2) - E^x^2 + 7*x +
x^2 - E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) +
6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^3, x] + 48*Defer[Int][E^(4 + 8*E^(4 - E^x^2) - E^x^2 + 8*x + x^2
- E^(4*x)*x^2 - E^(8*(E^(4 - E^x^2) + x))*x^2 + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)
*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 6 \exp \left (-e^{x^2}+4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (1-e^{2 e^{4-e^{x^2}}+x}\right )^3 x \left (-8 e^{4+2 e^{4-e^{x^2}}+x+x^2} x^2-e^{e^{x^2}} (1+2 x)+e^{2 e^{4-e^{x^2}}+e^{x^2}+x} (1+4 x)\right ) \, dx\\ &=6 \int \exp \left (-e^{x^2}+4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (1-e^{2 e^{4-e^{x^2}}+x}\right )^3 x \left (-8 e^{4+2 e^{4-e^{x^2}}+x+x^2} x^2-e^{e^{x^2}} (1+2 x)+e^{2 e^{4-e^{x^2}}+e^{x^2}+x} (1+4 x)\right ) \, dx\\ &=6 \int \left (8 \exp \left (4+2 e^{4-e^{x^2}}-e^{x^2}+5 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^3 x^3-\exp \left (4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^3 x \left (-1+e^{2 e^{4-e^{x^2}}+x}-2 x+4 e^{2 e^{4-e^{x^2}}+x} x\right )\right ) \, dx\\ &=-\left (6 \int \exp \left (4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^3 x \left (-1+e^{2 e^{4-e^{x^2}}+x}-2 x+4 e^{2 e^{4-e^{x^2}}+x} x\right ) \, dx\right )+48 \int \exp \left (4+2 e^{4-e^{x^2}}-e^{x^2}+5 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2\right ) \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^3 x^3 \, dx\\ &=-\left (6 \int \left (e^{4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+2 x)+6 e^{4 e^{4-e^{x^2}}+6 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+3 x)+e^{8 e^{4-e^{x^2}}+8 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+4 x)-2 e^{2 e^{4-e^{x^2}}+5 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (2+5 x)-2 e^{6 e^{4-e^{x^2}}+7 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (2+7 x)\right ) \, dx\right )+48 \int \left (-e^{4+2 e^{4-e^{x^2}}-e^{x^2}+5 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3+3 e^{4+4 e^{4-e^{x^2}}-e^{x^2}+6 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3-3 e^{4+6 e^{4-e^{x^2}}-e^{x^2}+7 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3+e^{4+8 e^{4-e^{x^2}}-e^{x^2}+8 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3\right ) \, dx\\ &=-\left (6 \int e^{4 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+2 x) \, dx\right )-6 \int e^{8 e^{4-e^{x^2}}+8 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+4 x) \, dx+12 \int e^{2 e^{4-e^{x^2}}+5 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (2+5 x) \, dx+12 \int e^{6 e^{4-e^{x^2}}+7 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (2+7 x) \, dx-36 \int e^{4 e^{4-e^{x^2}}+6 x-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x (1+3 x) \, dx-48 \int e^{4+2 e^{4-e^{x^2}}-e^{x^2}+5 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3 \, dx+48 \int e^{4+8 e^{4-e^{x^2}}-e^{x^2}+8 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3 \, dx+144 \int e^{4+4 e^{4-e^{x^2}}-e^{x^2}+6 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3 \, dx-144 \int e^{4+6 e^{4-e^{x^2}}-e^{x^2}+7 x+x^2-e^{4 x} x^2-e^{8 \left (e^{4-e^{x^2}}+x\right )} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2} x^3 \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.53, size = 35, normalized size = 1.03 \begin {gather*} 3 e^{-e^{4 x} \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4
- E^x^2) + 7*x)*x^2 - E^(8*E^(4 - E^x^2) + 8*x)*x^2)*(E^(4*x)*(-6*x - 12*x^2) + E^(8*E^(4 - E^x^2) + 8*x)*(-6*
x - 24*x^2 + 48*E^(4 - E^x^2 + x^2)*x^3) + E^(4*E^(4 - E^x^2) + 4*x)*(144*E^(4 - E^x^2 + 2*x + x^2)*x^3 + E^(2
*x)*(-36*x - 108*x^2)) + E^(2*E^(4 - E^x^2) + 2*x)*(-48*E^(4 - E^x^2 + 3*x + x^2)*x^3 + E^(3*x)*(24*x + 60*x^2
)) + E^(6*E^(4 - E^x^2) + 6*x)*(-144*E^(4 - E^x^2 + x + x^2)*x^3 + E^x*(24*x + 84*x^2))),x]

[Out]

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

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fricas [B]  time = 0.87, size = 216, normalized size = 6.35 \begin {gather*} 3 \, e^{\left (-{\left (x^{2} e^{\left (4 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )}\right )} - 4 \, x^{2} e^{\left (3 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 4 \, x\right )} + 6 \, x^{2} e^{\left (2 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 8 \, x\right )} - 4 \, x^{2} e^{\left ({\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 12 \, x\right )} + x^{2} e^{\left (16 \, x\right )}\right )} e^{\left (-12 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)
*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))
+(-108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2)*exp(4-exp(x^2))+(60*x^2+24*
x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp
(x)*exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*exp(x)^3*exp(2*exp(4-exp(x^
2))+2*x)+x^2*exp(x)^4),x, algorithm="fricas")

[Out]

3*e^(-(x^2*e^(4*(5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4))*e^(-x^2 - 3*x)) - 4*x^2*e^(3*(5*x*e^(x^2 +
3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4))*e^(-x^2 - 3*x) + 4*x) + 6*x^2*e^(2*(5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x
- e^(x^2) + 4))*e^(-x^2 - 3*x) + 8*x) - 4*x^2*e^((5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4))*e^(-x^2 -
3*x) + 12*x) + x^2*e^(16*x))*e^(-12*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -6 \, {\left ({\left (2 \, x^{2} + x\right )} e^{\left (4 \, x\right )} - {\left (8 \, x^{3} e^{\left (x^{2} - e^{\left (x^{2}\right )} + 4\right )} - 4 \, x^{2} - x\right )} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 2 \, {\left (12 \, x^{3} e^{\left (x^{2} + x - e^{\left (x^{2}\right )} + 4\right )} - {\left (7 \, x^{2} + 2 \, x\right )} e^{x}\right )} e^{\left (6 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, {\left (4 \, x^{3} e^{\left (x^{2} + 2 \, x - e^{\left (x^{2}\right )} + 4\right )} - {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x\right )}\right )} e^{\left (4 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 2 \, {\left (4 \, x^{3} e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )} - {\left (5 \, x^{2} + 2 \, x\right )} e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )} e^{\left (-x^{2} e^{\left (4 \, x\right )} - x^{2} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (7 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, x^{2} e^{\left (6 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (5 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)
*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))
+(-108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2)*exp(4-exp(x^2))+(60*x^2+24*
x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp
(x)*exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*exp(x)^3*exp(2*exp(4-exp(x^
2))+2*x)+x^2*exp(x)^4),x, algorithm="giac")

[Out]

integrate(-6*((2*x^2 + x)*e^(4*x) - (8*x^3*e^(x^2 - e^(x^2) + 4) - 4*x^2 - x)*e^(8*x + 8*e^(-e^(x^2) + 4)) + 2
*(12*x^3*e^(x^2 + x - e^(x^2) + 4) - (7*x^2 + 2*x)*e^x)*e^(6*x + 6*e^(-e^(x^2) + 4)) - 6*(4*x^3*e^(x^2 + 2*x -
e^(x^2) + 4) - (3*x^2 + x)*e^(2*x))*e^(4*x + 4*e^(-e^(x^2) + 4)) + 2*(4*x^3*e^(x^2 + 3*x - e^(x^2) + 4) - (5*
x^2 + 2*x)*e^(3*x))*e^(2*x + 2*e^(-e^(x^2) + 4)))*e^(-x^2*e^(4*x) - x^2*e^(8*x + 8*e^(-e^(x^2) + 4)) + 4*x^2*e
^(7*x + 6*e^(-e^(x^2) + 4)) - 6*x^2*e^(6*x + 4*e^(-e^(x^2) + 4)) + 4*x^2*e^(5*x + 2*e^(-e^(x^2) + 4))), x)

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maple [B]  time = 0.20, size = 84, normalized size = 2.47

method	result	size
risch	\(3 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{8 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}+8 x}+{\mathrm e}^{4 x}-4 \,{\mathrm e}^{7 x +6 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}+6 \,{\mathrm e}^{6 x +4 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}-4 \,{\mathrm e}^{5 x +2 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}\right )}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4
-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+(-108
*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2)*exp(4-exp(x^2))+(60*x^2+24*x)*exp
(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)*ex
p(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*exp(x)^3*exp(2*exp(4-exp(x^2))+2*
x)+x^2*exp(x)^4),x,method=_RETURNVERBOSE)

[Out]

3*exp(-x^2*(exp(8*exp(4-exp(x^2))+8*x)+exp(4*x)-4*exp(7*x+6*exp(4-exp(x^2)))+6*exp(6*x+4*exp(4-exp(x^2)))-4*ex
p(5*x+2*exp(4-exp(x^2)))))

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maxima [B]  time = 1.07, size = 97, normalized size = 2.85 \begin {gather*} 3 \, e^{\left (-x^{2} e^{\left (4 \, x\right )} - x^{2} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (7 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, x^{2} e^{\left (6 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (5 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)
*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))
+(-108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2)*exp(4-exp(x^2))+(60*x^2+24*
x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp
(x)*exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*exp(x)^3*exp(2*exp(4-exp(x^
2))+2*x)+x^2*exp(x)^4),x, algorithm="maxima")

[Out]

3*e^(-x^2*e^(4*x) - x^2*e^(8*x + 8*e^(-e^(x^2) + 4)) + 4*x^2*e^(7*x + 6*e^(-e^(x^2) + 4)) - 6*x^2*e^(6*x + 4*e
^(-e^(x^2) + 4)) + 4*x^2*e^(5*x + 2*e^(-e^(x^2) + 4)))

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mupad [B]  time = 1.86, size = 100, normalized size = 2.94 \begin {gather*} 3\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{5\,x}}\,{\mathrm {e}}^{-6\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{6\,x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{7\,x}}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{8\,x}}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{4\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(4*x^2*exp(6*x + 6*exp(4 - exp(x^2)))*exp(x) - x^2*exp(8*x + 8*exp(4 - exp(x^2))) - x^2*exp(4*x) + 4*x
^2*exp(3*x)*exp(2*x + 2*exp(4 - exp(x^2))) - 6*x^2*exp(2*x)*exp(4*x + 4*exp(4 - exp(x^2))))*(exp(4*x)*(6*x + 1
2*x^2) - exp(2*x + 2*exp(4 - exp(x^2)))*(exp(3*x)*(24*x + 60*x^2) - 48*x^3*exp(3*x)*exp(x^2)*exp(4 - exp(x^2))
) + exp(4*x + 4*exp(4 - exp(x^2)))*(exp(2*x)*(36*x + 108*x^2) - 144*x^3*exp(2*x)*exp(x^2)*exp(4 - exp(x^2))) -
exp(6*x + 6*exp(4 - exp(x^2)))*(exp(x)*(24*x + 84*x^2) - 144*x^3*exp(x^2)*exp(4 - exp(x^2))*exp(x)) + exp(8*x
+ 8*exp(4 - exp(x^2)))*(6*x + 24*x^2 - 48*x^3*exp(x^2)*exp(4 - exp(x^2)))),x)

[Out]

3*exp(4*x^2*exp(2*exp(-exp(x^2))*exp(4))*exp(5*x))*exp(-6*x^2*exp(4*exp(-exp(x^2))*exp(4))*exp(6*x))*exp(4*x^2
*exp(6*exp(-exp(x^2))*exp(4))*exp(7*x))*exp(-x^2*exp(8*exp(-exp(x^2))*exp(4))*exp(8*x))*exp(-x^2*exp(4*x))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x**3*exp(x**2)*exp(4-exp(x**2))-24*x**2-6*x)*exp(2*exp(4-exp(x**2))+2*x)**4+(-144*x**3*exp(x)*e
xp(x**2)*exp(4-exp(x**2))+(84*x**2+24*x)*exp(x))*exp(2*exp(4-exp(x**2))+2*x)**3+(144*x**3*exp(x)**2*exp(x**2)*
exp(4-exp(x**2))+(-108*x**2-36*x)*exp(x)**2)*exp(2*exp(4-exp(x**2))+2*x)**2+(-48*x**3*exp(x)**3*exp(x**2)*exp(
4-exp(x**2))+(60*x**2+24*x)*exp(x)**3)*exp(2*exp(4-exp(x**2))+2*x)+(-12*x**2-6*x)*exp(x)**4)/exp(x**2*exp(2*ex
p(4-exp(x**2))+2*x)**4-4*x**2*exp(x)*exp(2*exp(4-exp(x**2))+2*x)**3+6*x**2*exp(x)**2*exp(2*exp(4-exp(x**2))+2*
x)**2-4*x**2*exp(x)**3*exp(2*exp(4-exp(x**2))+2*x)+x**2*exp(x)**4),x)

[Out]

Timed out

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