3.9.5 \(\int e^{-2 x+e^{-2 x} (25 x^2+e^x (-10 x^2-10 x^3-10 e^4 x^3)+e^{2 x} (x^2+2 x^3+x^4+e^8 x^4+e^4 (2 x^3+2 x^4)))} (50 x-50 x^2+e^{2 x} (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 (6 x^2+8 x^3))+e^x (-20 x-20 x^2+10 x^3+e^4 (-30 x^2+10 x^3))) \, dx\)

Optimal. Leaf size=23 \[ e^{x^2 \left (1-5 e^{-x}+x+e^4 x\right )^2} \]

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Rubi [F]  time = 10.34, 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 (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-2*x + (25*x^2 + E^x*(-10*x^2 - 10*x^3 - 10*E^4*x^3) + E^(2*x)*(x^2 + 2*x^3 + x^4 + E^8*x^4 + E^4*(2*x^
3 + 2*x^4)))/E^(2*x))*(50*x - 50*x^2 + E^(2*x)*(2*x + 6*x^2 + 4*x^3 + 4*E^8*x^3 + E^4*(6*x^2 + 8*x^3)) + E^x*(
-20*x - 20*x^2 + 10*x^3 + E^4*(-30*x^2 + 10*x^3))),x]

[Out]

50*Defer[Int][E^(-2*x + ((-5 + E^x)^2*x^2)/E^(2*x) + (2*(1 + E^4)*(-5 + E^x)*x^3)/E^x + (1 + E^4)^2*x^4)*x, x]
- 20*Defer[Int][E^(-x + ((-5 + E^x)^2*x^2)/E^(2*x) + (2*(1 + E^4)*(-5 + E^x)*x^3)/E^x + (1 + E^4)^2*x^4)*x, x
] + 2*Defer[Int][E^((x^2*(-5 + E^(4 + x)*x + E^x*(1 + x))^2)/E^(2*x))*x, x] - 50*Defer[Int][E^(-2*x + ((-5 + E
^x)^2*x^2)/E^(2*x) + (2*(1 + E^4)*(-5 + E^x)*x^3)/E^x + (1 + E^4)^2*x^4)*x^2, x] - 10*(2 + 3*E^4)*Defer[Int][E
^(-x + ((-5 + E^x)^2*x^2)/E^(2*x) + (2*(1 + E^4)*(-5 + E^x)*x^3)/E^x + (1 + E^4)^2*x^4)*x^2, x] + 6*(1 + E^4)*
Defer[Int][E^((x^2*(-5 + E^(4 + x)*x + E^x*(1 + x))^2)/E^(2*x))*x^2, x] + 10*(1 + E^4)*Defer[Int][E^(-x + ((-5
+ E^x)^2*x^2)/E^(2*x) + (2*(1 + E^4)*(-5 + E^x)*x^3)/E^x + (1 + E^4)^2*x^4)*x^3, x] + 4*(1 + E^4)^2*Defer[Int
][E^((x^2*(-5 + E^(4 + x)*x + E^x*(1 + x))^2)/E^(2*x))*x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (50 \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x-50 \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x^2+10 \exp \left (-x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right )+2 \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right )\right ) \, dx\\ &=2 \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right ) \, dx+10 \int \exp \left (-x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x^2 \, dx\\ &=2 \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right ) \, dx+10 \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx\\ &=2 \int \left (\exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x+3 \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) \left (1+e^4\right ) x^2+2 \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) \left (1+e^4\right )^2 x^3\right ) \, dx+10 \int \left (-2 \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x-\exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) \left (2+3 e^4\right ) x^2+\exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) \left (1+e^4\right ) x^3\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx\\ &=2 \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x \, dx-20 \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx+\left (6 \left (1+e^4\right )\right ) \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x^2 \, dx+\left (10 \left (1+e^4\right )\right ) \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^3 \, dx+\left (4 \left (1+e^4\right )^2\right ) \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x^3 \, dx-\left (10 \left (2+3 e^4\right )\right ) \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 29, normalized size = 1.26 \begin {gather*} e^{e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-2*x + (25*x^2 + E^x*(-10*x^2 - 10*x^3 - 10*E^4*x^3) + E^(2*x)*(x^2 + 2*x^3 + x^4 + E^8*x^4 + E^4
*(2*x^3 + 2*x^4)))/E^(2*x))*(50*x - 50*x^2 + E^(2*x)*(2*x + 6*x^2 + 4*x^3 + 4*E^8*x^3 + E^4*(6*x^2 + 8*x^3)) +
E^x*(-20*x - 20*x^2 + 10*x^3 + E^4*(-30*x^2 + 10*x^3))),x]

[Out]

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

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fricas [B]  time = 0.98, size = 70, normalized size = 3.04 \begin {gather*} e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4} - 2 \, x\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^
2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4*exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10
*x^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm="fricas")

[Out]

e^((25*x^2 + (x^4*e^8 + x^4 + 2*x^3 + x^2 + 2*(x^4 + x^3)*e^4 - 2*x)*e^(2*x) - 10*(x^3*e^4 + x^3 + x^2)*e^x)*e
^(-2*x) + 2*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -2 \, {\left (25 \, x^{2} - {\left (2 \, x^{3} e^{8} + 2 \, x^{3} + 3 \, x^{2} + {\left (4 \, x^{3} + 3 \, x^{2}\right )} e^{4} + x\right )} e^{\left (2 \, x\right )} - 5 \, {\left (x^{3} - 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{4} - 2 \, x\right )} e^{x} - 25 \, x\right )} e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^
2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4*exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10
*x^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm="giac")

[Out]

integrate(-2*(25*x^2 - (2*x^3*e^8 + 2*x^3 + 3*x^2 + (4*x^3 + 3*x^2)*e^4 + x)*e^(2*x) - 5*(x^3 - 2*x^2 + (x^3 -
3*x^2)*e^4 - 2*x)*e^x - 25*x)*e^((25*x^2 + (x^4*e^8 + x^4 + 2*x^3 + x^2 + 2*(x^4 + x^3)*e^4)*e^(2*x) - 10*(x^
3*e^4 + x^3 + x^2)*e^x)*e^(-2*x) - 2*x), x)

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maple [B]  time = 0.41, size = 73, normalized size = 3.17

method	result	size
norman	\({\mathrm e}^{\left (\left (x^{4} {\mathrm e}^{8}+\left (2 x^{4}+2 x^{3}\right ) {\mathrm e}^{4}+x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-10 x^{3} {\mathrm e}^{4}-10 x^{3}-10 x^{2}\right ) {\mathrm e}^{x}+25 x^{2}\right ) {\mathrm e}^{-2 x}}\)	\(73\)
risch	\({\mathrm e}^{-x^{2} \left (-2 x^{2} {\mathrm e}^{2 x +4}-x^{2} {\mathrm e}^{2 x +8}+10 x \,{\mathrm e}^{4+x}-2 x \,{\mathrm e}^{2 x +4}-{\mathrm e}^{2 x} x^{2}+10 \,{\mathrm e}^{x} x -2 x \,{\mathrm e}^{2 x}+10 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-25\right ) {\mathrm e}^{-2 x}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^2-20*x
)*exp(x)-50*x^2+50*x)*exp(((x^4*exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10*x^3-1
0*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 2.03, size = 71, normalized size = 3.09 \begin {gather*} e^{\left (x^{4} e^{8} + 2 \, x^{4} e^{4} + x^{4} + 2 \, x^{3} e^{4} - 10 \, x^{3} e^{\left (-x\right )} - 10 \, x^{3} e^{\left (-x + 4\right )} + 2 \, x^{3} - 10 \, x^{2} e^{\left (-x\right )} + 25 \, x^{2} e^{\left (-2 \, x\right )} + x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3*exp(4)^2+(8*x^3+6*x^2)*exp(4)+4*x^3+6*x^2+2*x)*exp(x)^2+((10*x^3-30*x^2)*exp(4)+10*x^3-20*x^
2-20*x)*exp(x)-50*x^2+50*x)*exp(((x^4*exp(4)^2+(2*x^4+2*x^3)*exp(4)+x^4+2*x^3+x^2)*exp(x)^2+(-10*x^3*exp(4)-10
*x^3-10*x^2)*exp(x)+25*x^2)/exp(x)^2)/exp(x)^2,x, algorithm="maxima")

[Out]

e^(x^4*e^8 + 2*x^4*e^4 + x^4 + 2*x^3*e^4 - 10*x^3*e^(-x) - 10*x^3*e^(-x + 4) + 2*x^3 - 10*x^2*e^(-x) + 25*x^2*
e^(-2*x) + x^2)

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mupad [B]  time = 1.29, size = 80, normalized size = 3.48 \begin {gather*} {\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^8}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-10\,x^2\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{25\,x^2\,{\mathrm {e}}^{-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2*x)*exp(exp(-2*x)*(exp(2*x)*(exp(4)*(2*x^3 + 2*x^4) + x^4*exp(8) + x^2 + 2*x^3 + x^4) - exp(x)*(10*x
^3*exp(4) + 10*x^2 + 10*x^3) + 25*x^2))*(50*x + exp(2*x)*(2*x + exp(4)*(6*x^2 + 8*x^3) + 4*x^3*exp(8) + 6*x^2
+ 4*x^3) - exp(x)*(20*x + exp(4)*(30*x^2 - 10*x^3) + 20*x^2 - 10*x^3) - 50*x^2),x)

[Out]

exp(2*x^3*exp(4))*exp(2*x^4*exp(4))*exp(x^4*exp(8))*exp(x^2)*exp(x^4)*exp(-10*x^3*exp(-x)*exp(4))*exp(2*x^3)*e
xp(-10*x^2*exp(-x))*exp(-10*x^3*exp(-x))*exp(25*x^2*exp(-2*x))

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sympy [B]  time = 0.50, size = 71, normalized size = 3.09 \begin {gather*} e^{\left (25 x^{2} + \left (- 10 x^{3} e^{4} - 10 x^{3} - 10 x^{2}\right ) e^{x} + \left (x^{4} + x^{4} e^{8} + 2 x^{3} + x^{2} + \left (2 x^{4} + 2 x^{3}\right ) e^{4}\right ) e^{2 x}\right ) e^{- 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3*exp(4)**2+(8*x**3+6*x**2)*exp(4)+4*x**3+6*x**2+2*x)*exp(x)**2+((10*x**3-30*x**2)*exp(4)+10*
x**3-20*x**2-20*x)*exp(x)-50*x**2+50*x)*exp(((x**4*exp(4)**2+(2*x**4+2*x**3)*exp(4)+x**4+2*x**3+x**2)*exp(x)**
2+(-10*x**3*exp(4)-10*x**3-10*x**2)*exp(x)+25*x**2)/exp(x)**2)/exp(x)**2,x)

[Out]

exp((25*x**2 + (-10*x**3*exp(4) - 10*x**3 - 10*x**2)*exp(x) + (x**4 + x**4*exp(8) + 2*x**3 + x**2 + (2*x**4 +
2*x**3)*exp(4))*exp(2*x))*exp(-2*x))

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