3.89.52 \(\int \frac {e^{-e^x x^2+\frac {e^{-e^x x^2} (4 e^{x+e^x x^2}+x^2)}{4 x}} (5 x^2+e^x (-10 x^4-5 x^5)+e^{e^x x^2} (-20 x+e^x (-20+20 x)))}{4 x^3} \, dx\)

Optimal. Leaf size=32 \[ \frac {5 e^{\frac {e^x+\frac {1}{4} e^{-e^x x^2} x^2}{x}}}{x} \]

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Rubi [F]  time = 6.33, 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 x^2+\frac {e^{-e^x x^2} \left (4 e^{x+e^x x^2}+x^2\right )}{4 x}\right ) \left (5 x^2+e^x \left (-10 x^4-5 x^5\right )+e^{e^x x^2} \left (-20 x+e^x (-20+20 x)\right )\right )}{4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.18, size = 30, normalized size = 0.94 \begin {gather*} \frac {5 e^{\frac {e^x}{x}+\frac {1}{4} e^{-e^x x^2} x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-(E^x*x^2) + (4*E^(x + E^x*x^2) + x^2)/(4*E^(E^x*x^2)*x))*(5*x^2 + E^x*(-10*x^4 - 5*x^5) + E^(E^
x*x^2)*(-20*x + E^x*(-20 + 20*x))))/(4*x^3),x]

[Out]

(5*E^(E^x/x + x/(4*E^(E^x*x^2))))/x

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fricas [A]  time = 0.48, size = 55, normalized size = 1.72 \begin {gather*} \frac {5 \, e^{\left (x^{2} e^{x} + \frac {{\left (x^{2} e^{x} - 4 \, {\left (x^{3} - 1\right )} e^{\left (x^{2} e^{x} + 2 \, x\right )}\right )} e^{\left (-x^{2} e^{x} - x\right )}}{4 \, x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((20*x-20)*exp(x)-20*x)*exp(exp(x)*x^2)+(-5*x^5-10*x^4)*exp(x)+5*x^2)*exp(1/4*(4*exp(x)*exp(exp
(x)*x^2)+x^2)/x/exp(exp(x)*x^2))/x^3/exp(exp(x)*x^2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 \, {\left (x^{2} + 4 \, {\left ({\left (x - 1\right )} e^{x} - x\right )} e^{\left (x^{2} e^{x}\right )} - {\left (x^{5} + 2 \, x^{4}\right )} e^{x}\right )} e^{\left (-x^{2} e^{x} + \frac {{\left (x^{2} + 4 \, e^{\left (x^{2} e^{x} + x\right )}\right )} e^{\left (-x^{2} e^{x}\right )}}{4 \, x}\right )}}{4 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((20*x-20)*exp(x)-20*x)*exp(exp(x)*x^2)+(-5*x^5-10*x^4)*exp(x)+5*x^2)*exp(1/4*(4*exp(x)*exp(exp
(x)*x^2)+x^2)/x/exp(exp(x)*x^2))/x^3/exp(exp(x)*x^2),x, algorithm="giac")

[Out]

integrate(5/4*(x^2 + 4*((x - 1)*e^x - x)*e^(x^2*e^x) - (x^5 + 2*x^4)*e^x)*e^(-x^2*e^x + 1/4*(x^2 + 4*e^(x^2*e^
x + x))*e^(-x^2*e^x)/x)/x^3, x)

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maple [A]  time = 0.08, size = 35, normalized size = 1.09

method	result	size
risch	\(\frac {5 \,{\mathrm e}^{\frac {\left (4 \,{\mathrm e}^{x \left ({\mathrm e}^{x} x +1\right )}+x^{2}\right ) {\mathrm e}^{-{\mathrm e}^{x} x^{2}}}{4 x}}}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(((20*x-20)*exp(x)-20*x)*exp(exp(x)*x^2)+(-5*x^5-10*x^4)*exp(x)+5*x^2)*exp(1/4*(4*exp(x)*exp(exp(x)*x^
2)+x^2)/x/exp(exp(x)*x^2))/x^3/exp(exp(x)*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.54, size = 24, normalized size = 0.75 \begin {gather*} \frac {5 \, e^{\left (\frac {1}{4} \, x e^{\left (-x^{2} e^{x}\right )} + \frac {e^{x}}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((20*x-20)*exp(x)-20*x)*exp(exp(x)*x^2)+(-5*x^5-10*x^4)*exp(x)+5*x^2)*exp(1/4*(4*exp(x)*exp(exp
(x)*x^2)+x^2)/x/exp(exp(x)*x^2))/x^3/exp(exp(x)*x^2),x, algorithm="maxima")

[Out]

5*e^(1/4*x*e^(-x^2*e^x) + e^x/x)/x

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mupad [B]  time = 5.29, size = 24, normalized size = 0.75 \begin {gather*} \frac {5\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}+\frac {x\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(-x^2*exp(x))*(exp(x^2*exp(x))*exp(x) + x^2/4))/x)*exp(-x^2*exp(x))*(exp(x)*(10*x^4 + 5*x^5) + e
xp(x^2*exp(x))*(20*x - exp(x)*(20*x - 20)) - 5*x^2))/(4*x^3),x)

[Out]

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

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sympy [A]  time = 0.52, size = 31, normalized size = 0.97 \begin {gather*} \frac {5 e^{\frac {\left (\frac {x^{2}}{4} + e^{x} e^{x^{2} e^{x}}\right ) e^{- x^{2} e^{x}}}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(((20*x-20)*exp(x)-20*x)*exp(exp(x)*x**2)+(-5*x**5-10*x**4)*exp(x)+5*x**2)*exp(1/4*(4*exp(x)*exp
(exp(x)*x**2)+x**2)/x/exp(exp(x)*x**2))/x**3/exp(exp(x)*x**2),x)

[Out]

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

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