3.98.79 \(\int \frac {2 x^3+e^{\frac {1}{2} (-e^{\frac {1}{x^2}+e^x x}+e^{\frac {1}{x^2}} (3+x))} (e^{\frac {1}{x^2}} (-6-2 x+x^3)+e^{e^x x} (2 e^{\frac {1}{x^2}}+e^{\frac {1}{x^2}+x} (-x^3-x^4)))}{2 x^3} \, dx\)

Optimal. Leaf size=31 \[ -3-e^3+e^{\frac {1}{2} e^{\frac {1}{x^2}} \left (3-e^{e^x x}+x\right )}+x \]

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Rubi [F]  time = 11.14, 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 {2 x^3+\exp \left (\frac {1}{2} \left (-e^{\frac {1}{x^2}+e^x x}+e^{\frac {1}{x^2}} (3+x)\right )\right ) \left (e^{\frac {1}{x^2}} \left (-6-2 x+x^3\right )+e^{e^x x} \left (2 e^{\frac {1}{x^2}}+e^{\frac {1}{x^2}+x} \left (-x^3-x^4\right )\right )\right )}{2 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.66, size = 32, normalized size = 1.03 \begin {gather*} e^{-\frac {1}{2} e^{\frac {1}{x^2}+e^x x}+\frac {1}{2} e^{\frac {1}{x^2}} (3+x)}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 41, normalized size = 1.32 \begin {gather*} x + e^{\left (\frac {1}{2} \, {\left ({\left (x + 3\right )} e^{\left (\frac {x^{3} + 1}{x^{2}}\right )} - e^{\left (x + \frac {x^{3} e^{x} + 1}{x^{2}}\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((((-x^4-x^3)*exp(1/x^2)*exp(x)+2*exp(1/x^2))*exp(exp(x)*x)+(x^3-2*x-6)*exp(1/x^2))*exp(-1/2*exp
(1/x^2)*exp(exp(x)*x)+1/2*(3+x)*exp(1/x^2))+2*x^3)/x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} - {\left ({\left ({\left (x^{4} + x^{3}\right )} e^{\left (x + \frac {1}{x^{2}}\right )} - 2 \, e^{\left (\frac {1}{x^{2}}\right )}\right )} e^{\left (x e^{x}\right )} - {\left (x^{3} - 2 \, x - 6\right )} e^{\left (\frac {1}{x^{2}}\right )}\right )} e^{\left (\frac {1}{2} \, {\left (x + 3\right )} e^{\left (\frac {1}{x^{2}}\right )} - \frac {1}{2} \, e^{\left (x e^{x} + \frac {1}{x^{2}}\right )}\right )}}{2 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((((-x^4-x^3)*exp(1/x^2)*exp(x)+2*exp(1/x^2))*exp(exp(x)*x)+(x^3-2*x-6)*exp(1/x^2))*exp(-1/2*exp
(1/x^2)*exp(exp(x)*x)+1/2*(3+x)*exp(1/x^2))+2*x^3)/x^3,x, algorithm="giac")

[Out]

integrate(1/2*(2*x^3 - (((x^4 + x^3)*e^(x + 1/x^2) - 2*e^(x^(-2)))*e^(x*e^x) - (x^3 - 2*x - 6)*e^(x^(-2)))*e^(
1/2*(x + 3)*e^(x^(-2)) - 1/2*e^(x*e^x + 1/x^2)))/x^3, x)

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maple [A]  time = 0.09, size = 33, normalized size = 1.06

method	result	size
risch	\(x +{\mathrm e}^{-\frac {{\mathrm e}^{\frac {1+{\mathrm e}^{x} x^{3}}{x^{2}}}}{2}+\frac {x \,{\mathrm e}^{\frac {1}{x^{2}}}}{2}+\frac {3 \,{\mathrm e}^{\frac {1}{x^{2}}}}{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((((-x^4-x^3)*exp(1/x^2)*exp(x)+2*exp(1/x^2))*exp(exp(x)*x)+(x^3-2*x-6)*exp(1/x^2))*exp(-1/2*exp(1/x^2
)*exp(exp(x)*x)+1/2*(3+x)*exp(1/x^2))+2*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 28, normalized size = 0.90 \begin {gather*} x + e^{\left (\frac {1}{2} \, x e^{\left (\frac {1}{x^{2}}\right )} - \frac {1}{2} \, e^{\left (x e^{x} + \frac {1}{x^{2}}\right )} + \frac {3}{2} \, e^{\left (\frac {1}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((((-x^4-x^3)*exp(1/x^2)*exp(x)+2*exp(1/x^2))*exp(exp(x)*x)+(x^3-2*x-6)*exp(1/x^2))*exp(-1/2*exp
(1/x^2)*exp(exp(x)*x)+1/2*(3+x)*exp(1/x^2))+2*x^3)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.76, size = 28, normalized size = 0.90 \begin {gather*} x+{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}+\frac {x\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}-\frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {1}{x^2}}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp((exp(1/x^2)*(x + 3))/2 - (exp(x*exp(x))*exp(1/x^2))/2)*(exp(1/x^2)*(2*x - x^3 + 6) - exp(x*exp(x))*
(2*exp(1/x^2) - exp(1/x^2)*exp(x)*(x^3 + x^4))))/2 - x^3)/x^3,x)

[Out]

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

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sympy [A]  time = 12.05, size = 31, normalized size = 1.00 \begin {gather*} x + e^{\left (\frac {x}{2} + \frac {3}{2}\right ) e^{\frac {1}{x^{2}}} - \frac {e^{\frac {1}{x^{2}}} e^{x e^{x}}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((((-x**4-x**3)*exp(1/x**2)*exp(x)+2*exp(1/x**2))*exp(exp(x)*x)+(x**3-2*x-6)*exp(1/x**2))*exp(-1
/2*exp(1/x**2)*exp(exp(x)*x)+1/2*(3+x)*exp(1/x**2))+2*x**3)/x**3,x)

[Out]

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

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