3.75.43 \(\int (-1+32 x-24 x^2+4 x^3+e^{2 x^2} (-8+66 x-32 x^2+4 x^3)+e^{x^2} (32-32 x+70 x^2-32 x^3+4 x^4)+e^{3 e^{2 x}-3 x} (e^{2 x^2} (24-48 e^{2 x}-32 x)-16 x+24 x^2-48 e^{2 x} x^2+e^{x^2} (-16+48 x-96 e^{2 x} x-32 x^2))+e^{4 e^{2 x}-4 x} (2 x-4 x^2+8 e^{2 x} x^2+e^{2 x^2} (-4+8 e^{2 x}+4 x)+e^{x^2} (2-8 x+16 e^{2 x} x+4 x^2))+e^{2 e^{2 x}-2 x} (48 x-54 x^2+4 x^3+e^{2 x^2} (-50+e^{2 x} (96-8 x)+100 x-8 x^2)+e^{2 x} (96 x^2-8 x^3)+e^{x^2} (48-104 x+104 x^2-8 x^3+e^{2 x} (192 x-16 x^2)))+e^{e^{2 x}-x} (-64 x+56 x^2-8 x^3+e^{2 x} (-64 x^2+16 x^3)+e^{2 x^2} (40-136 x+32 x^2+e^{2 x} (-64+16 x))+e^{x^2} (-64+96 x-144 x^2+32 x^3+e^{2 x} (-128 x+32 x^2)))) \, dx\)

Optimal. Leaf size=36 \[ -2-x+\left (e^{x^2}+x\right )^2 \left (-\left (-2+e^{e^{2 x}-x}\right )^2+x\right )^2 \]

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Rubi [F]  time = 14.20, 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 \left (-1+32 x-24 x^2+4 x^3+e^{2 x^2} \left (-8+66 x-32 x^2+4 x^3\right )+e^{x^2} \left (32-32 x+70 x^2-32 x^3+4 x^4\right )+e^{3 e^{2 x}-3 x} \left (e^{2 x^2} \left (24-48 e^{2 x}-32 x\right )-16 x+24 x^2-48 e^{2 x} x^2+e^{x^2} \left (-16+48 x-96 e^{2 x} x-32 x^2\right )\right )+e^{4 e^{2 x}-4 x} \left (2 x-4 x^2+8 e^{2 x} x^2+e^{2 x^2} \left (-4+8 e^{2 x}+4 x\right )+e^{x^2} \left (2-8 x+16 e^{2 x} x+4 x^2\right )\right )+e^{2 e^{2 x}-2 x} \left (48 x-54 x^2+4 x^3+e^{2 x^2} \left (-50+e^{2 x} (96-8 x)+100 x-8 x^2\right )+e^{2 x} \left (96 x^2-8 x^3\right )+e^{x^2} \left (48-104 x+104 x^2-8 x^3+e^{2 x} \left (192 x-16 x^2\right )\right )\right )+e^{e^{2 x}-x} \left (-64 x+56 x^2-8 x^3+e^{2 x} \left (-64 x^2+16 x^3\right )+e^{2 x^2} \left (40-136 x+32 x^2+e^{2 x} (-64+16 x)\right )+e^{x^2} \left (-64+96 x-144 x^2+32 x^3+e^{2 x} \left (-128 x+32 x^2\right )\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[-1 + 32*x - 24*x^2 + 4*x^3 + E^(2*x^2)*(-8 + 66*x - 32*x^2 + 4*x^3) + E^x^2*(32 - 32*x + 70*x^2 - 32*x^3 +
4*x^4) + E^(3*E^(2*x) - 3*x)*(E^(2*x^2)*(24 - 48*E^(2*x) - 32*x) - 16*x + 24*x^2 - 48*E^(2*x)*x^2 + E^x^2*(-1
6 + 48*x - 96*E^(2*x)*x - 32*x^2)) + E^(4*E^(2*x) - 4*x)*(2*x - 4*x^2 + 8*E^(2*x)*x^2 + E^(2*x^2)*(-4 + 8*E^(2
*x) + 4*x) + E^x^2*(2 - 8*x + 16*E^(2*x)*x + 4*x^2)) + E^(2*E^(2*x) - 2*x)*(48*x - 54*x^2 + 4*x^3 + E^(2*x^2)*
(-50 + E^(2*x)*(96 - 8*x) + 100*x - 8*x^2) + E^(2*x)*(96*x^2 - 8*x^3) + E^x^2*(48 - 104*x + 104*x^2 - 8*x^3 +
E^(2*x)*(192*x - 16*x^2))) + E^(E^(2*x) - x)*(-64*x + 56*x^2 - 8*x^3 + E^(2*x)*(-64*x^2 + 16*x^3) + E^(2*x^2)*
(40 - 136*x + 32*x^2 + E^(2*x)*(-64 + 16*x)) + E^x^2*(-64 + 96*x - 144*x^2 + 32*x^3 + E^(2*x)*(-128*x + 32*x^2
))),x]

[Out]

16*E^(2*x^2) - x + 32*E^x^2*x - 8*E^(2*x^2)*x + 16*x^2 - 16*E^x^2*x^2 + E^(2*x^2)*x^2 - 8*x^3 + 2*E^x^2*x^3 +
x^4 + 96*Defer[Int][E^(2*(E^(2*x) + x^2)), x] + 48*Defer[Int][E^(2*(E^(2*x) - x) + x^2), x] - 16*Defer[Int][E^
(3*(E^(2*x) - x) + x^2), x] + 2*Defer[Int][E^(4*(E^(2*x) - x) + x^2), x] - 4*Defer[Int][E^(2*(2*E^(2*x) - 2*x
+ x^2)), x] - 64*Defer[Int][E^(E^(2*x) - x + x^2), x] - 50*Defer[Int][E^(2*(E^(2*x) - x + x^2)), x] + 8*Defer[
Int][E^(2*(2*E^(2*x) - x + x^2)), x] + 24*Defer[Int][E^(3*(E^(2*x) - x) + 2*x^2), x] + 40*Defer[Int][E^(E^(2*x
) - x + 2*x^2), x] - 64*Defer[Int][E^(E^(2*x) + x + 2*x^2), x] - 48*Defer[Int][E^(3*(E^(2*x) - x) + 2*x + 2*x^
2), x] - 64*Defer[Int][E^(E^(2*x) - x)*x, x] + 48*Defer[Int][E^(2*(E^(2*x) - x))*x, x] - 16*Defer[Int][E^(3*(E
^(2*x) - x))*x, x] + 2*Defer[Int][E^(4*(E^(2*x) - x))*x, x] - 8*Defer[Int][E^(2*(E^(2*x) + x^2))*x, x] + 192*D
efer[Int][E^(2*E^(2*x) + x^2)*x, x] - 104*Defer[Int][E^(2*(E^(2*x) - x) + x^2)*x, x] + 48*Defer[Int][E^(3*(E^(
2*x) - x) + x^2)*x, x] - 8*Defer[Int][E^(4*(E^(2*x) - x) + x^2)*x, x] + 4*Defer[Int][E^(2*(2*E^(2*x) - 2*x + x
^2))*x, x] + 96*Defer[Int][E^(E^(2*x) - x + x^2)*x, x] + 100*Defer[Int][E^(2*(E^(2*x) - x + x^2))*x, x] - 128*
Defer[Int][E^(E^(2*x) + x + x^2)*x, x] - 96*Defer[Int][E^(3*(E^(2*x) - x) + 2*x + x^2)*x, x] + 16*Defer[Int][E
^(4*(E^(2*x) - x) + 2*x + x^2)*x, x] - 32*Defer[Int][E^(3*(E^(2*x) - x) + 2*x^2)*x, x] - 136*Defer[Int][E^(E^(
2*x) - x + 2*x^2)*x, x] + 16*Defer[Int][E^(E^(2*x) + x + 2*x^2)*x, x] + 96*Defer[Int][E^(2*E^(2*x))*x^2, x] +
56*Defer[Int][E^(E^(2*x) - x)*x^2, x] - 54*Defer[Int][E^(2*(E^(2*x) - x))*x^2, x] + 24*Defer[Int][E^(3*(E^(2*x
) - x))*x^2, x] - 4*Defer[Int][E^(4*(E^(2*x) - x))*x^2, x] + 8*Defer[Int][E^(2*(2*E^(2*x) - x))*x^2, x] - 64*D
efer[Int][E^(E^(2*x) + x)*x^2, x] - 48*Defer[Int][E^(3*(E^(2*x) - x) + 2*x)*x^2, x] - 16*Defer[Int][E^(2*E^(2*
x) + x^2)*x^2, x] + 104*Defer[Int][E^(2*(E^(2*x) - x) + x^2)*x^2, x] - 32*Defer[Int][E^(3*(E^(2*x) - x) + x^2)
*x^2, x] + 4*Defer[Int][E^(4*(E^(2*x) - x) + x^2)*x^2, x] - 144*Defer[Int][E^(E^(2*x) - x + x^2)*x^2, x] - 8*D
efer[Int][E^(2*(E^(2*x) - x + x^2))*x^2, x] + 32*Defer[Int][E^(E^(2*x) + x + x^2)*x^2, x] + 32*Defer[Int][E^(E
^(2*x) - x + 2*x^2)*x^2, x] - 8*Defer[Int][E^(2*E^(2*x))*x^3, x] - 8*Defer[Int][E^(E^(2*x) - x)*x^3, x] + 4*De
fer[Int][E^(2*(E^(2*x) - x))*x^3, x] + 16*Defer[Int][E^(E^(2*x) + x)*x^3, x] - 8*Defer[Int][E^(2*(E^(2*x) - x)
+ x^2)*x^3, x] + 32*Defer[Int][E^(E^(2*x) - x + x^2)*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x+16 x^2-8 x^3+x^4+\int e^{2 x^2} \left (-8+66 x-32 x^2+4 x^3\right ) \, dx+\int e^{x^2} \left (32-32 x+70 x^2-32 x^3+4 x^4\right ) \, dx+\int e^{3 e^{2 x}-3 x} \left (e^{2 x^2} \left (24-48 e^{2 x}-32 x\right )-16 x+24 x^2-48 e^{2 x} x^2+e^{x^2} \left (-16+48 x-96 e^{2 x} x-32 x^2\right )\right ) \, dx+\int e^{4 e^{2 x}-4 x} \left (2 x-4 x^2+8 e^{2 x} x^2+e^{2 x^2} \left (-4+8 e^{2 x}+4 x\right )+e^{x^2} \left (2-8 x+16 e^{2 x} x+4 x^2\right )\right ) \, dx+\int e^{2 e^{2 x}-2 x} \left (48 x-54 x^2+4 x^3+e^{2 x^2} \left (-50+e^{2 x} (96-8 x)+100 x-8 x^2\right )+e^{2 x} \left (96 x^2-8 x^3\right )+e^{x^2} \left (48-104 x+104 x^2-8 x^3+e^{2 x} \left (192 x-16 x^2\right )\right )\right ) \, dx+\int e^{e^{2 x}-x} \left (-64 x+56 x^2-8 x^3+e^{2 x} \left (-64 x^2+16 x^3\right )+e^{2 x^2} \left (40-136 x+32 x^2+e^{2 x} (-64+16 x)\right )+e^{x^2} \left (-64+96 x-144 x^2+32 x^3+e^{2 x} \left (-128 x+32 x^2\right )\right )\right ) \, dx\\ &=-x+16 x^2-8 x^3+x^4+\int 2 e^{4 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (1+4 e^{x (2+x)}+2 e^{x^2} (-1+x)-2 x+4 e^{2 x} x\right ) \, dx+\int \left (-8 e^{2 x^2}+66 e^{2 x^2} x-32 e^{2 x^2} x^2+4 e^{2 x^2} x^3\right ) \, dx+\int \left (32 e^{x^2}-32 e^{x^2} x+70 e^{x^2} x^2-32 e^{x^2} x^3+4 e^{x^2} x^4\right ) \, dx+\int 8 e^{3 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (-2-6 e^{x (2+x)}+3 x-6 e^{2 x} x-e^{x^2} (-3+4 x)\right ) \, dx+\int 2 e^{2 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (24-4 e^{x (2+x)} (-12+x)-27 x-4 e^{2 x} (-12+x) x+2 x^2-e^{x^2} \left (25-50 x+4 x^2\right )\right ) \, dx+\int 8 e^{e^{2 x}-x} \left (e^{x^2}+x\right ) \left (-8+2 e^{x (2+x)} (-4+x)+7 x+2 e^{2 x} (-4+x) x-x^2+e^{x^2} \left (5-17 x+4 x^2\right )\right ) \, dx\\ &=-x+16 x^2-8 x^3+x^4+2 \int e^{4 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (1+4 e^{x (2+x)}+2 e^{x^2} (-1+x)-2 x+4 e^{2 x} x\right ) \, dx+2 \int e^{2 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (24-4 e^{x (2+x)} (-12+x)-27 x-4 e^{2 x} (-12+x) x+2 x^2-e^{x^2} \left (25-50 x+4 x^2\right )\right ) \, dx+4 \int e^{2 x^2} x^3 \, dx+4 \int e^{x^2} x^4 \, dx-8 \int e^{2 x^2} \, dx+8 \int e^{3 \left (e^{2 x}-x\right )} \left (e^{x^2}+x\right ) \left (-2-6 e^{x (2+x)}+3 x-6 e^{2 x} x-e^{x^2} (-3+4 x)\right ) \, dx+8 \int e^{e^{2 x}-x} \left (e^{x^2}+x\right ) \left (-8+2 e^{x (2+x)} (-4+x)+7 x+2 e^{2 x} (-4+x) x-x^2+e^{x^2} \left (5-17 x+4 x^2\right )\right ) \, dx+32 \int e^{x^2} \, dx-32 \int e^{x^2} x \, dx-32 \int e^{2 x^2} x^2 \, dx-32 \int e^{x^2} x^3 \, dx+66 \int e^{2 x^2} x \, dx+70 \int e^{x^2} x^2 \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 9.35, size = 289, normalized size = 8.03 \begin {gather*} 16 e^{2 x^2}-x+32 e^{x^2} x-8 e^{2 x^2} x+16 x^2-16 e^{x^2} x^2+e^{2 x^2} x^2-8 x^3+2 e^{x^2} x^3+x^4+2 e^{4 e^{2 x}} \left (\frac {1}{2} e^{-4 x+2 x^2}+e^{-4 x+x^2} x+\frac {1}{2} e^{-4 x} x^2\right )-8 e^{3 e^{2 x}} \left (e^{-3 x+2 x^2}+2 e^{-3 x+x^2} x+e^{-3 x} x^2\right )-2 e^{2 e^{2 x}} \left (e^{-2 x+2 x^2} (-12+x)+e^{-2 x+x^2} \left (-24 x+2 x^2\right )+e^{-2 x} \left (-12 x^2+x^3\right )\right )+8 e^{e^{2 x}} \left (e^{-x+2 x^2} (-4+x)+e^{-x+x^2} \left (-8 x+2 x^2\right )+e^{-x} \left (-4 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 32*x - 24*x^2 + 4*x^3 + E^(2*x^2)*(-8 + 66*x - 32*x^2 + 4*x^3) + E^x^2*(32 - 32*x + 70*x^2 - 32
*x^3 + 4*x^4) + E^(3*E^(2*x) - 3*x)*(E^(2*x^2)*(24 - 48*E^(2*x) - 32*x) - 16*x + 24*x^2 - 48*E^(2*x)*x^2 + E^x
^2*(-16 + 48*x - 96*E^(2*x)*x - 32*x^2)) + E^(4*E^(2*x) - 4*x)*(2*x - 4*x^2 + 8*E^(2*x)*x^2 + E^(2*x^2)*(-4 +
8*E^(2*x) + 4*x) + E^x^2*(2 - 8*x + 16*E^(2*x)*x + 4*x^2)) + E^(2*E^(2*x) - 2*x)*(48*x - 54*x^2 + 4*x^3 + E^(2
*x^2)*(-50 + E^(2*x)*(96 - 8*x) + 100*x - 8*x^2) + E^(2*x)*(96*x^2 - 8*x^3) + E^x^2*(48 - 104*x + 104*x^2 - 8*
x^3 + E^(2*x)*(192*x - 16*x^2))) + E^(E^(2*x) - x)*(-64*x + 56*x^2 - 8*x^3 + E^(2*x)*(-64*x^2 + 16*x^3) + E^(2
*x^2)*(40 - 136*x + 32*x^2 + E^(2*x)*(-64 + 16*x)) + E^x^2*(-64 + 96*x - 144*x^2 + 32*x^3 + E^(2*x)*(-128*x +
32*x^2))),x]

[Out]

16*E^(2*x^2) - x + 32*E^x^2*x - 8*E^(2*x^2)*x + 16*x^2 - 16*E^x^2*x^2 + E^(2*x^2)*x^2 - 8*x^3 + 2*E^x^2*x^3 +
x^4 + 2*E^(4*E^(2*x))*(E^(-4*x + 2*x^2)/2 + E^(-4*x + x^2)*x + x^2/(2*E^(4*x))) - 8*E^(3*E^(2*x))*(E^(-3*x + 2
*x^2) + 2*E^(-3*x + x^2)*x + x^2/E^(3*x)) - 2*E^(2*E^(2*x))*(E^(-2*x + 2*x^2)*(-12 + x) + E^(-2*x + x^2)*(-24*
x + 2*x^2) + (-12*x^2 + x^3)/E^(2*x)) + 8*E^E^(2*x)*(E^(-x + 2*x^2)*(-4 + x) + E^(-x + x^2)*(-8*x + 2*x^2) + (
-4*x^2 + x^3)/E^x)

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fricas [B]  time = 1.12, size = 197, normalized size = 5.47 \begin {gather*} x^{4} - 8 \, x^{3} + 16 \, x^{2} + {\left (x^{2} - 8 \, x + 16\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} e^{\left (x^{2}\right )} + 8 \, {\left (x^{3} - 4 \, x^{2} + {\left (x - 4\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-x + e^{\left (2 \, x\right )}\right )} - 2 \, {\left (x^{3} - 12 \, x^{2} + {\left (x - 12\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{2} - 12 \, x\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 8 \, {\left (x^{2} + 2 \, x e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-3 \, x + 3 \, e^{\left (2 \, x\right )}\right )} + {\left (x^{2} + 2 \, x e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-4 \, x + 4 \, e^{\left (2 \, x\right )}\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+4*x-4)*exp(x^2)^2+(16*x*exp(x)^2+4*x^2-8*x+2)*exp(x^2)+8*exp(x)^2*x^2-4*x^2+2*x)*exp(ex
p(x)^2-x)^4+((-48*exp(x)^2-32*x+24)*exp(x^2)^2+(-96*x*exp(x)^2-32*x^2+48*x-16)*exp(x^2)-48*exp(x)^2*x^2+24*x^2
-16*x)*exp(exp(x)^2-x)^3+(((-8*x+96)*exp(x)^2-8*x^2+100*x-50)*exp(x^2)^2+((-16*x^2+192*x)*exp(x)^2-8*x^3+104*x
^2-104*x+48)*exp(x^2)+(-8*x^3+96*x^2)*exp(x)^2+4*x^3-54*x^2+48*x)*exp(exp(x)^2-x)^2+(((16*x-64)*exp(x)^2+32*x^
2-136*x+40)*exp(x^2)^2+((32*x^2-128*x)*exp(x)^2+32*x^3-144*x^2+96*x-64)*exp(x^2)+(16*x^3-64*x^2)*exp(x)^2-8*x^
3+56*x^2-64*x)*exp(exp(x)^2-x)+(4*x^3-32*x^2+66*x-8)*exp(x^2)^2+(4*x^4-32*x^3+70*x^2-32*x+32)*exp(x^2)+4*x^3-2
4*x^2+32*x-1,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.23, size = 383, normalized size = 10.64 \begin {gather*} x^{4} - 8 \, x^{3} + 16 \, x^{2} + {\left (x^{2} - 8 \, x + 16\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} e^{\left (x^{2}\right )} - 8 \, {\left (x^{2} e^{\left (-2 \, x + 6 \, e^{\left (2 \, x\right )}\right )} + 2 \, x e^{\left (x^{2} - 2 \, x + 6 \, e^{\left (2 \, x\right )}\right )} + e^{\left (2 \, x^{2} - 2 \, x + 6 \, e^{\left (2 \, x\right )}\right )}\right )} e^{\left (-x - 3 \, e^{\left (2 \, x\right )}\right )} - 2 \, {\left (x^{3} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} + 2 \, x^{2} e^{\left (x^{2} + 4 \, e^{\left (2 \, x\right )}\right )} - 12 \, x^{2} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} + x e^{\left (2 \, x^{2} + 4 \, e^{\left (2 \, x\right )}\right )} - 24 \, x e^{\left (x^{2} + 4 \, e^{\left (2 \, x\right )}\right )} - 12 \, e^{\left (2 \, x^{2} + 4 \, e^{\left (2 \, x\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{\left (2 \, x\right )}\right )} + 8 \, {\left (x^{3} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} + 2 \, x^{2} e^{\left (x^{2} + 2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 4 \, x^{2} e^{\left (2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} + x e^{\left (2 \, x^{2} + 2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 8 \, x e^{\left (x^{2} + 2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 4 \, e^{\left (2 \, x^{2} + 2 \, x + 2 \, e^{\left (2 \, x\right )}\right )}\right )} e^{\left (-3 \, x - e^{\left (2 \, x\right )}\right )} + {\left (x^{2} e^{\left (-4 \, x + 8 \, e^{\left (2 \, x\right )}\right )} + 2 \, x e^{\left (x^{2} - 4 \, x + 8 \, e^{\left (2 \, x\right )}\right )} + e^{\left (2 \, x^{2} - 4 \, x + 8 \, e^{\left (2 \, x\right )}\right )}\right )} e^{\left (-4 \, e^{\left (2 \, x\right )}\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+4*x-4)*exp(x^2)^2+(16*x*exp(x)^2+4*x^2-8*x+2)*exp(x^2)+8*exp(x)^2*x^2-4*x^2+2*x)*exp(ex
p(x)^2-x)^4+((-48*exp(x)^2-32*x+24)*exp(x^2)^2+(-96*x*exp(x)^2-32*x^2+48*x-16)*exp(x^2)-48*exp(x)^2*x^2+24*x^2
-16*x)*exp(exp(x)^2-x)^3+(((-8*x+96)*exp(x)^2-8*x^2+100*x-50)*exp(x^2)^2+((-16*x^2+192*x)*exp(x)^2-8*x^3+104*x
^2-104*x+48)*exp(x^2)+(-8*x^3+96*x^2)*exp(x)^2+4*x^3-54*x^2+48*x)*exp(exp(x)^2-x)^2+(((16*x-64)*exp(x)^2+32*x^
2-136*x+40)*exp(x^2)^2+((32*x^2-128*x)*exp(x)^2+32*x^3-144*x^2+96*x-64)*exp(x^2)+(16*x^3-64*x^2)*exp(x)^2-8*x^
3+56*x^2-64*x)*exp(exp(x)^2-x)+(4*x^3-32*x^2+66*x-8)*exp(x^2)^2+(4*x^4-32*x^3+70*x^2-32*x+32)*exp(x^2)+4*x^3-2
4*x^2+32*x-1,x, algorithm="giac")

[Out]

x^4 - 8*x^3 + 16*x^2 + (x^2 - 8*x + 16)*e^(2*x^2) + 2*(x^3 - 8*x^2 + 16*x)*e^(x^2) - 8*(x^2*e^(-2*x + 6*e^(2*x
)) + 2*x*e^(x^2 - 2*x + 6*e^(2*x)) + e^(2*x^2 - 2*x + 6*e^(2*x)))*e^(-x - 3*e^(2*x)) - 2*(x^3*e^(4*e^(2*x)) +
2*x^2*e^(x^2 + 4*e^(2*x)) - 12*x^2*e^(4*e^(2*x)) + x*e^(2*x^2 + 4*e^(2*x)) - 24*x*e^(x^2 + 4*e^(2*x)) - 12*e^(
2*x^2 + 4*e^(2*x)))*e^(-2*x - 2*e^(2*x)) + 8*(x^3*e^(2*x + 2*e^(2*x)) + 2*x^2*e^(x^2 + 2*x + 2*e^(2*x)) - 4*x^
2*e^(2*x + 2*e^(2*x)) + x*e^(2*x^2 + 2*x + 2*e^(2*x)) - 8*x*e^(x^2 + 2*x + 2*e^(2*x)) - 4*e^(2*x^2 + 2*x + 2*e
^(2*x)))*e^(-3*x - e^(2*x)) + (x^2*e^(-4*x + 8*e^(2*x)) + 2*x*e^(x^2 - 4*x + 8*e^(2*x)) + e^(2*x^2 - 4*x + 8*e
^(2*x)))*e^(-4*e^(2*x)) - x

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maple [B]  time = 0.17, size = 224, normalized size = 6.22

method	result	size
risch	\(\left (x^{2}+2 \,{\mathrm e}^{x^{2}} x +{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{4 \,{\mathrm e}^{2 x}-4 x}+\left (-8 x^{2}-16 \,{\mathrm e}^{x^{2}} x -8 \,{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{3 \,{\mathrm e}^{2 x}-3 x}+\left (-2 x^{3}-4 x^{2} {\mathrm e}^{x^{2}}-2 x \,{\mathrm e}^{2 x^{2}}+24 x^{2}+48 \,{\mathrm e}^{x^{2}} x +24 \,{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{2 \,{\mathrm e}^{2 x}-2 x}+\left (8 x^{3}+16 x^{2} {\mathrm e}^{x^{2}}+8 x \,{\mathrm e}^{2 x^{2}}-32 x^{2}-64 \,{\mathrm e}^{x^{2}} x -32 \,{\mathrm e}^{2 x^{2}}\right ) {\mathrm e}^{{\mathrm e}^{2 x}-x}+\left (x^{2}-8 x +16\right ) {\mathrm e}^{2 x^{2}}+\left (2 x^{3}-16 x^{2}+32 x \right ) {\mathrm e}^{x^{2}}+x^{4}-8 x^{3}+16 x^{2}-x\)	\(224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(x)^2+4*x-4)*exp(x^2)^2+(16*x*exp(x)^2+4*x^2-8*x+2)*exp(x^2)+8*exp(x)^2*x^2-4*x^2+2*x)*exp(exp(x)^2
-x)^4+((-48*exp(x)^2-32*x+24)*exp(x^2)^2+(-96*x*exp(x)^2-32*x^2+48*x-16)*exp(x^2)-48*exp(x)^2*x^2+24*x^2-16*x)
*exp(exp(x)^2-x)^3+(((-8*x+96)*exp(x)^2-8*x^2+100*x-50)*exp(x^2)^2+((-16*x^2+192*x)*exp(x)^2-8*x^3+104*x^2-104
*x+48)*exp(x^2)+(-8*x^3+96*x^2)*exp(x)^2+4*x^3-54*x^2+48*x)*exp(exp(x)^2-x)^2+(((16*x-64)*exp(x)^2+32*x^2-136*
x+40)*exp(x^2)^2+((32*x^2-128*x)*exp(x)^2+32*x^3-144*x^2+96*x-64)*exp(x^2)+(16*x^3-64*x^2)*exp(x)^2-8*x^3+56*x
^2-64*x)*exp(exp(x)^2-x)+(4*x^3-32*x^2+66*x-8)*exp(x^2)^2+(4*x^4-32*x^3+70*x^2-32*x+32)*exp(x^2)+4*x^3-24*x^2+
32*x-1,x,method=_RETURNVERBOSE)

[Out]

(x^2+2*exp(x^2)*x+exp(2*x^2))*exp(4*exp(2*x)-4*x)+(-8*x^2-16*exp(x^2)*x-8*exp(2*x^2))*exp(3*exp(2*x)-3*x)+(-2*
x^3-4*x^2*exp(x^2)-2*x*exp(2*x^2)+24*x^2+48*exp(x^2)*x+24*exp(2*x^2))*exp(2*exp(2*x)-2*x)+(8*x^3+16*x^2*exp(x^
2)+8*x*exp(2*x^2)-32*x^2-64*exp(x^2)*x-32*exp(2*x^2))*exp(exp(2*x)-x)+(x^2-8*x+16)*exp(2*x^2)+(2*x^3-16*x^2+32
*x)*exp(x^2)+x^4-8*x^3+16*x^2-x

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maxima [B]  time = 0.41, size = 197, normalized size = 5.47 \begin {gather*} x^{4} - 8 \, x^{3} + 16 \, x^{2} + {\left (x^{2} - 8 \, x + 16\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} e^{\left (x^{2}\right )} + 8 \, {\left (x^{3} - 4 \, x^{2} + {\left (x - 4\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{2} - 4 \, x\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-x + e^{\left (2 \, x\right )}\right )} - 2 \, {\left (x^{3} - 12 \, x^{2} + {\left (x - 12\right )} e^{\left (2 \, x^{2}\right )} + 2 \, {\left (x^{2} - 12 \, x\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-2 \, x + 2 \, e^{\left (2 \, x\right )}\right )} - 8 \, {\left (x^{2} + 2 \, x e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-3 \, x + 3 \, e^{\left (2 \, x\right )}\right )} + {\left (x^{2} + 2 \, x e^{\left (x^{2}\right )} + e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-4 \, x + 4 \, e^{\left (2 \, x\right )}\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)^2+4*x-4)*exp(x^2)^2+(16*x*exp(x)^2+4*x^2-8*x+2)*exp(x^2)+8*exp(x)^2*x^2-4*x^2+2*x)*exp(ex
p(x)^2-x)^4+((-48*exp(x)^2-32*x+24)*exp(x^2)^2+(-96*x*exp(x)^2-32*x^2+48*x-16)*exp(x^2)-48*exp(x)^2*x^2+24*x^2
-16*x)*exp(exp(x)^2-x)^3+(((-8*x+96)*exp(x)^2-8*x^2+100*x-50)*exp(x^2)^2+((-16*x^2+192*x)*exp(x)^2-8*x^3+104*x
^2-104*x+48)*exp(x^2)+(-8*x^3+96*x^2)*exp(x)^2+4*x^3-54*x^2+48*x)*exp(exp(x)^2-x)^2+(((16*x-64)*exp(x)^2+32*x^
2-136*x+40)*exp(x^2)^2+((32*x^2-128*x)*exp(x)^2+32*x^3-144*x^2+96*x-64)*exp(x^2)+(16*x^3-64*x^2)*exp(x)^2-8*x^
3+56*x^2-64*x)*exp(exp(x)^2-x)+(4*x^3-32*x^2+66*x-8)*exp(x^2)^2+(4*x^4-32*x^3+70*x^2-32*x+32)*exp(x^2)+4*x^3-2
4*x^2+32*x-1,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.36, size = 225, normalized size = 6.25 \begin {gather*} {\mathrm {e}}^{2\,x^2}\,\left (x^2-8\,x+16\right )-x-{\mathrm {e}}^{3\,{\mathrm {e}}^{2\,x}-3\,x}\,\left (8\,{\mathrm {e}}^{2\,x^2}+16\,x\,{\mathrm {e}}^{x^2}+8\,x^2\right )+{\mathrm {e}}^{x^2}\,\left (2\,x^3-16\,x^2+32\,x\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-2\,x}\,\left (24\,{\mathrm {e}}^{2\,x^2}+48\,x\,{\mathrm {e}}^{x^2}-2\,x\,{\mathrm {e}}^{2\,x^2}-4\,x^2\,{\mathrm {e}}^{x^2}+24\,x^2-2\,x^3\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}-4\,x}\,\left ({\mathrm {e}}^{2\,x^2}+2\,x\,{\mathrm {e}}^{x^2}+x^2\right )+16\,x^2-8\,x^3+x^4-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}-x}\,\left (32\,{\mathrm {e}}^{2\,x^2}+64\,x\,{\mathrm {e}}^{x^2}-8\,x\,{\mathrm {e}}^{2\,x^2}-16\,x^2\,{\mathrm {e}}^{x^2}+32\,x^2-8\,x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(32*x + exp(2*exp(2*x) - 2*x)*(48*x - exp(2*x^2)*(exp(2*x)*(8*x - 96) - 100*x + 8*x^2 + 50) + exp(x^2)*(exp
(2*x)*(192*x - 16*x^2) - 104*x + 104*x^2 - 8*x^3 + 48) + exp(2*x)*(96*x^2 - 8*x^3) - 54*x^2 + 4*x^3) + exp(4*e
xp(2*x) - 4*x)*(2*x + exp(x^2)*(16*x*exp(2*x) - 8*x + 4*x^2 + 2) + 8*x^2*exp(2*x) + exp(2*x^2)*(4*x + 8*exp(2*
x) - 4) - 4*x^2) - exp(3*exp(2*x) - 3*x)*(16*x + exp(x^2)*(96*x*exp(2*x) - 48*x + 32*x^2 + 16) + 48*x^2*exp(2*
x) + exp(2*x^2)*(32*x + 48*exp(2*x) - 24) - 24*x^2) + exp(x^2)*(70*x^2 - 32*x - 32*x^3 + 4*x^4 + 32) + exp(2*x
^2)*(66*x - 32*x^2 + 4*x^3 - 8) - exp(exp(2*x) - x)*(64*x - exp(2*x^2)*(exp(2*x)*(16*x - 64) - 136*x + 32*x^2
+ 40) + exp(x^2)*(exp(2*x)*(128*x - 32*x^2) - 96*x + 144*x^2 - 32*x^3 + 64) + exp(2*x)*(64*x^2 - 16*x^3) - 56*
x^2 + 8*x^3) - 24*x^2 + 4*x^3 - 1,x)

[Out]

exp(2*x^2)*(x^2 - 8*x + 16) - x - exp(3*exp(2*x) - 3*x)*(8*exp(2*x^2) + 16*x*exp(x^2) + 8*x^2) + exp(x^2)*(32*
x - 16*x^2 + 2*x^3) + exp(2*exp(2*x) - 2*x)*(24*exp(2*x^2) + 48*x*exp(x^2) - 2*x*exp(2*x^2) - 4*x^2*exp(x^2) +
24*x^2 - 2*x^3) + exp(4*exp(2*x) - 4*x)*(exp(2*x^2) + 2*x*exp(x^2) + x^2) + 16*x^2 - 8*x^3 + x^4 - exp(exp(2*
x) - x)*(32*exp(2*x^2) + 64*x*exp(x^2) - 8*x*exp(2*x^2) - 16*x^2*exp(x^2) + 32*x^2 - 8*x^3)

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sympy [B]  time = 6.66, size = 230, normalized size = 6.39 \begin {gather*} x^{4} - 8 x^{3} + 16 x^{2} - x + \left (- 8 x^{2} - 16 x e^{x^{2}} - 8 e^{2 x^{2}}\right ) e^{- 3 x + 3 e^{2 x}} + \left (x^{2} - 8 x + 16\right ) e^{2 x^{2}} + \left (x^{2} + 2 x e^{x^{2}} + e^{2 x^{2}}\right ) e^{- 4 x + 4 e^{2 x}} + \left (2 x^{3} - 16 x^{2} + 32 x\right ) e^{x^{2}} + \left (- 2 x^{3} - 4 x^{2} e^{x^{2}} + 24 x^{2} - 2 x e^{2 x^{2}} + 48 x e^{x^{2}} + 24 e^{2 x^{2}}\right ) e^{- 2 x + 2 e^{2 x}} + \left (8 x^{3} + 16 x^{2} e^{x^{2}} - 32 x^{2} + 8 x e^{2 x^{2}} - 64 x e^{x^{2}} - 32 e^{2 x^{2}}\right ) e^{- x + e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(x)**2+4*x-4)*exp(x**2)**2+(16*x*exp(x)**2+4*x**2-8*x+2)*exp(x**2)+8*exp(x)**2*x**2-4*x**2+2*
x)*exp(exp(x)**2-x)**4+((-48*exp(x)**2-32*x+24)*exp(x**2)**2+(-96*x*exp(x)**2-32*x**2+48*x-16)*exp(x**2)-48*ex
p(x)**2*x**2+24*x**2-16*x)*exp(exp(x)**2-x)**3+(((-8*x+96)*exp(x)**2-8*x**2+100*x-50)*exp(x**2)**2+((-16*x**2+
192*x)*exp(x)**2-8*x**3+104*x**2-104*x+48)*exp(x**2)+(-8*x**3+96*x**2)*exp(x)**2+4*x**3-54*x**2+48*x)*exp(exp(
x)**2-x)**2+(((16*x-64)*exp(x)**2+32*x**2-136*x+40)*exp(x**2)**2+((32*x**2-128*x)*exp(x)**2+32*x**3-144*x**2+9
6*x-64)*exp(x**2)+(16*x**3-64*x**2)*exp(x)**2-8*x**3+56*x**2-64*x)*exp(exp(x)**2-x)+(4*x**3-32*x**2+66*x-8)*ex
p(x**2)**2+(4*x**4-32*x**3+70*x**2-32*x+32)*exp(x**2)+4*x**3-24*x**2+32*x-1,x)

[Out]

x**4 - 8*x**3 + 16*x**2 - x + (-8*x**2 - 16*x*exp(x**2) - 8*exp(2*x**2))*exp(-3*x + 3*exp(2*x)) + (x**2 - 8*x
+ 16)*exp(2*x**2) + (x**2 + 2*x*exp(x**2) + exp(2*x**2))*exp(-4*x + 4*exp(2*x)) + (2*x**3 - 16*x**2 + 32*x)*ex
p(x**2) + (-2*x**3 - 4*x**2*exp(x**2) + 24*x**2 - 2*x*exp(2*x**2) + 48*x*exp(x**2) + 24*exp(2*x**2))*exp(-2*x
+ 2*exp(2*x)) + (8*x**3 + 16*x**2*exp(x**2) - 32*x**2 + 8*x*exp(2*x**2) - 64*x*exp(x**2) - 32*exp(2*x**2))*exp
(-x + exp(2*x))

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