3.28.48 \(\int \frac {e^{e^{-x^2} (e^{x^2} (-5+x)-3 x)-x-x^2} (12 x-12 x^2-24 x^3+24 x^4+e^x (3 x^2-3 x^3-6 x^4+6 x^5)+e^{x^2} (4-12 x+e^x (-3 x^2+x^3)))}{-x^2+3 x^3-3 x^4+x^5} \, dx\) [2748]

Optimal. Leaf size=33 \[ \frac {e^{-5+x-3 e^{-x^2} x} \left (4 e^{-x}+x\right )}{(-1+x)^2 x} \]

[Out]

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

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Rubi [F]
time = 6.82, 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 {\exp \left (e^{-x^2} \left (e^{x^2} (-5+x)-3 x\right )-x-x^2\right ) \left (12 x-12 x^2-24 x^3+24 x^4+e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )+e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{-x^2+3 x^3-3 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((E^x^2*(-5 + x) - 3*x)/E^x^2 - x - x^2)*(12*x - 12*x^2 - 24*x^3 + 24*x^4 + E^x*(3*x^2 - 3*x^3 - 6*x^4
+ 6*x^5) + E^x^2*(4 - 12*x + E^x*(-3*x^2 + x^3))))/(-x^2 + 3*x^3 - 3*x^4 + x^5),x]

[Out]

6*Defer[Int][E^(-5 + x - (3*x)/E^x^2 - x^2), x] - 8*Defer[Int][E^(-5 - (3*x)/E^x^2)/(-1 + x)^3, x] - 2*Defer[I
nt][E^(-5 + x - (3*x)/E^x^2)/(-1 + x)^3, x] + 4*Defer[Int][E^(-5 - (3*x)/E^x^2)/(-1 + x)^2, x] + Defer[Int][E^
(-5 + x - (3*x)/E^x^2)/(-1 + x)^2, x] + 12*Defer[Int][E^(-5 - (3*x)/E^x^2 - x^2)/(-1 + x)^2, x] + 3*Defer[Int]
[E^(-5 + x - (3*x)/E^x^2 - x^2)/(-1 + x)^2, x] + 36*Defer[Int][E^(-5 - (3*x)/E^x^2 - x^2)/(-1 + x), x] + 12*De
fer[Int][E^(-5 + x - (3*x)/E^x^2 - x^2)/(-1 + x), x] - 4*Defer[Int][E^(-5 - (3*x)/E^x^2)/x^2, x] - 12*Defer[In
t][E^(-5 - (3*x)/E^x^2 - x^2)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} \left (-12 x+12 x^2+24 x^3-24 x^4-e^x \left (3 x^2-3 x^3-6 x^4+6 x^5\right )-e^{x^2} \left (4-12 x+e^x \left (-3 x^2+x^3\right )\right )\right )}{(1-x)^3 x^2} \, dx\\ &=\int \left (-\frac {12 e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )}}{(-1+x)^3}+\frac {12 e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )}}{(-1+x)^3 x}-\frac {24 e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} x}{(-1+x)^3}+\frac {24 e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} x^2}{(-1+x)^3}+\frac {3 e^{x-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} \left (-1+2 x^2\right )}{(-1+x)^2}+\frac {e^{x^2-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} \left (4-12 x-3 e^x x^2+e^x x^3\right )}{(-1+x)^3 x^2}\right ) \, dx\\ &=3 \int \frac {e^{x-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} \left (-1+2 x^2\right )}{(-1+x)^2} \, dx-12 \int \frac {e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )}}{(-1+x)^3} \, dx+12 \int \frac {e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )}}{(-1+x)^3 x} \, dx-24 \int \frac {e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} x}{(-1+x)^3} \, dx+24 \int \frac {e^{-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} x^2}{(-1+x)^3} \, dx+\int \frac {e^{x^2-e^{-x^2} \left (5 e^{x^2}+3 x+e^{x^2} x^2\right )} \left (4-12 x-3 e^x x^2+e^x x^3\right )}{(-1+x)^3 x^2} \, dx\\ &=3 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2} \left (-1+2 x^2\right )}{(1-x)^2} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3} \, dx+12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3 x} \, dx-24 \int \frac {e^{-5-3 e^{-x^2} x-x^2} x}{(-1+x)^3} \, dx+24 \int \frac {e^{-5-3 e^{-x^2} x-x^2} x^2}{(-1+x)^3} \, dx+\int \frac {e^{-5-3 e^{-x^2} x} \left (-4+12 x+3 e^x x^2-e^x x^3\right )}{(1-x)^3 x^2} \, dx\\ &=3 \int \left (2 e^{-5+x-3 e^{-x^2} x-x^2}+\frac {e^{-5+x-3 e^{-x^2} x-x^2}}{(-1+x)^2}+\frac {4 e^{-5+x-3 e^{-x^2} x-x^2}}{-1+x}\right ) \, dx+12 \int \left (\frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3}-\frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2}+\frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x}-\frac {e^{-5-3 e^{-x^2} x-x^2}}{x}\right ) \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3} \, dx-24 \int \left (\frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3}+\frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2}\right ) \, dx+24 \int \left (\frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^3}+\frac {2 e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2}+\frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x}\right ) \, dx+\int \left (\frac {e^{-5+x-3 e^{-x^2} x} (-3+x)}{(-1+x)^3}-\frac {4 e^{-5-3 e^{-x^2} x} (-1+3 x)}{(-1+x)^3 x^2}\right ) \, dx\\ &=3 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx-4 \int \frac {e^{-5-3 e^{-x^2} x} (-1+3 x)}{(-1+x)^3 x^2} \, dx+6 \int e^{-5+x-3 e^{-x^2} x-x^2} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+12 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{-1+x} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{x} \, dx-24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+48 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+\int \frac {e^{-5+x-3 e^{-x^2} x} (-3+x)}{(-1+x)^3} \, dx\\ &=3 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx-4 \int \left (\frac {2 e^{-5-3 e^{-x^2} x}}{(-1+x)^3}-\frac {e^{-5-3 e^{-x^2} x}}{(-1+x)^2}+\frac {e^{-5-3 e^{-x^2} x}}{x^2}\right ) \, dx+6 \int e^{-5+x-3 e^{-x^2} x-x^2} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+12 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{-1+x} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{x} \, dx-24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+48 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+\int \left (-\frac {2 e^{-5+x-3 e^{-x^2} x}}{(-1+x)^3}+\frac {e^{-5+x-3 e^{-x^2} x}}{(-1+x)^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-5+x-3 e^{-x^2} x}}{(-1+x)^3} \, dx\right )+3 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+4 \int \frac {e^{-5-3 e^{-x^2} x}}{(-1+x)^2} \, dx-4 \int \frac {e^{-5-3 e^{-x^2} x}}{x^2} \, dx+6 \int e^{-5+x-3 e^{-x^2} x-x^2} \, dx-8 \int \frac {e^{-5-3 e^{-x^2} x}}{(-1+x)^3} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+12 \int \frac {e^{-5+x-3 e^{-x^2} x-x^2}}{-1+x} \, dx-12 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{x} \, dx-24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+24 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{-1+x} \, dx+48 \int \frac {e^{-5-3 e^{-x^2} x-x^2}}{(-1+x)^2} \, dx+\int \frac {e^{-5+x-3 e^{-x^2} x}}{(-1+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.09, size = 30, normalized size = 0.91 \begin {gather*} \frac {e^{-5-3 e^{-x^2} x} \left (4+e^x x\right )}{(-1+x)^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E^x^2*(-5 + x) - 3*x)/E^x^2 - x - x^2)*(12*x - 12*x^2 - 24*x^3 + 24*x^4 + E^x*(3*x^2 - 3*x^3 -
6*x^4 + 6*x^5) + E^x^2*(4 - 12*x + E^x*(-3*x^2 + x^3))))/(-x^2 + 3*x^3 - 3*x^4 + x^5),x]

[Out]

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

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Maple [A]
time = 0.10, size = 59, normalized size = 1.79

method	result	size
risch	\(\frac {\left ({\mathrm e}^{x} x +4\right ) {\mathrm e}^{{\mathrm e}^{x^{2}} {\mathrm e}^{-x^{2}} x -5 \,{\mathrm e}^{x^{2}} {\mathrm e}^{-x^{2}}-3 \,{\mathrm e}^{-x^{2}} x -x}}{\left (x^{2}-2 x +1\right ) x}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)*exp(x)+24*x^4-24*x^3-12*x^2+12*x)*exp(((x-
5)*exp(x^2)-3*x)/exp(x^2))/(x^5-3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)*exp(x)+24*x^4-24*x^3-12*x^2+12*x)*ex
p(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5-3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm="maxima")

[Out]

integrate((24*x^4 - 24*x^3 - 12*x^2 + ((x^3 - 3*x^2)*e^x - 12*x + 4)*e^(x^2) + 3*(2*x^5 - 2*x^4 - x^3 + x^2)*e
^x + 12*x)*e^(-x^2 + ((x - 5)*e^(x^2) - 3*x)*e^(-x^2) - x)/(x^5 - 3*x^4 + 3*x^3 - x^2), x)

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Fricas [A]
time = 0.35, size = 46, normalized size = 1.39 \begin {gather*} \frac {{\left (x e^{x} + 4\right )} e^{\left (x^{2} - {\left ({\left (x^{2} + 5\right )} e^{\left (x^{2}\right )} + 3 \, x\right )} e^{\left (-x^{2}\right )}\right )}}{x^{3} - 2 \, x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)*exp(x)+24*x^4-24*x^3-12*x^2+12*x)*ex
p(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5-3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.23, size = 44, normalized size = 1.33 \begin {gather*} \frac {\left (x e^{x} + 4\right ) e^{\left (- 3 x + \left (x - 5\right ) e^{x^{2}}\right ) e^{- x^{2}}}}{x^{3} e^{x} - 2 x^{2} e^{x} + x e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-3*x**2)*exp(x)-12*x+4)*exp(x**2)+(6*x**5-6*x**4-3*x**3+3*x**2)*exp(x)+24*x**4-24*x**3-12*x**
2+12*x)*exp(((-5+x)*exp(x**2)-3*x)/exp(x**2))/(x**5-3*x**4+3*x**3-x**2)/exp(x)/exp(x**2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-3*x^2)*exp(x)-12*x+4)*exp(x^2)+(6*x^5-6*x^4-3*x^3+3*x^2)*exp(x)+24*x^4-24*x^3-12*x^2+12*x)*ex
p(((-5+x)*exp(x^2)-3*x)/exp(x^2))/(x^5-3*x^4+3*x^3-x^2)/exp(x)/exp(x^2),x, algorithm="giac")

[Out]

integrate((24*x^4 - 24*x^3 - 12*x^2 + ((x^3 - 3*x^2)*e^x - 12*x + 4)*e^(x^2) + 3*(2*x^5 - 2*x^4 - x^3 + x^2)*e
^x + 12*x)*e^(-x^2 + ((x - 5)*e^(x^2) - 3*x)*e^(-x^2) - x)/(x^5 - 3*x^4 + 3*x^3 - x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{-{\mathrm {e}}^{-x^2}\,\left (3\,x-{\mathrm {e}}^{x^2}\,\left (x-5\right )\right )}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-x^2}\,\left (12\,x+{\mathrm {e}}^x\,\left (6\,x^5-6\,x^4-3\,x^3+3\,x^2\right )-12\,x^2-24\,x^3+24\,x^4-{\mathrm {e}}^{x^2}\,\left (12\,x+{\mathrm {e}}^x\,\left (3\,x^2-x^3\right )-4\right )\right )}{-x^5+3\,x^4-3\,x^3+x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(-x^2)*(3*x - exp(x^2)*(x - 5)))*exp(-x)*exp(-x^2)*(12*x + exp(x)*(3*x^2 - 3*x^3 - 6*x^4 + 6*x^5
) - 12*x^2 - 24*x^3 + 24*x^4 - exp(x^2)*(12*x + exp(x)*(3*x^2 - x^3) - 4)))/(x^2 - 3*x^3 + 3*x^4 - x^5),x)

[Out]

int(-(exp(-exp(-x^2)*(3*x - exp(x^2)*(x - 5)))*exp(-x)*exp(-x^2)*(12*x + exp(x)*(3*x^2 - 3*x^3 - 6*x^4 + 6*x^5
) - 12*x^2 - 24*x^3 + 24*x^4 - exp(x^2)*(12*x + exp(x)*(3*x^2 - x^3) - 4)))/(x^2 - 3*x^3 + 3*x^4 - x^5), x)

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