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

Optimal. Leaf size=30 \[ e^{e^{e^{-3+\frac {5}{x}-2 x+\frac {-\frac {e^5}{x}+x}{e^5}}}} \]

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Rubi [F]  time = 5.69, 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 (-5+e^{e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}}+e^{\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}}+\frac {x^2+e^5 \left (4-3 x-2 x^2\right )}{e^5 x}\right ) \left (x^2+e^5 \left (-4-2 x^2\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(1 - 2*E^5)*Defer[Int][E^(-8 + E^E^(-3 + 4/x + (-2 + E^(-5))*x) + E^(-3 + 4/x + (-2 + E^(-5))*x) + 4/x - 2*(1
- 1/(2*E^5))*x), x] - 4*Defer[Int][E^(-3 + E^E^(-3 + 4/x + (-2 + E^(-5))*x) + E^(-3 + 4/x + (-2 + E^(-5))*x) +
4/x - 2*(1 - 1/(2*E^5))*x)/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 1.73, size = 20, normalized size = 0.67 \begin {gather*} e^{e^{e^{-3+\frac {4}{x}+\left (-2+\frac {1}{e^5}\right ) x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^E^E^(-3 + 4/x + (-2 + E^(-5))*x)

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fricas [B]  time = 0.87, size = 136, normalized size = 4.53 \begin {gather*} e^{\left (-\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x} + \frac {{\left (x^{2} - 2 \, {\left (x^{2} + 4 \, x - 2\right )} e^{5} + x e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x} + 5\right )} + x e^{\left (e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )} + 5\right )}\right )} e^{\left (-5\right )}}{x} - e^{\left (\frac {{\left (x^{2} - {\left (2 \, x^{2} + 3 \, x - 4\right )} e^{5}\right )} e^{\left (-5\right )}}{x}\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.43, size = 29, normalized size = 0.97

method	result	size
norman	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {\left (\left (-2 x^{2}-3 x +4\right ) {\mathrm e}^{5}+x^{2}\right ) {\mathrm e}^{-5}}{x}}}}\)	\(29\)
risch	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{-\frac {\left (2 x^{2} {\mathrm e}^{5}+3 x \,{\mathrm e}^{5}-x^{2}-4 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{x}}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-4)*exp(5)+x^2)*exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))*exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/ex
p(5)))*exp(exp(exp(((-2*x^2-3*x+4)*exp(5)+x^2)/x/exp(5))))/x^2/exp(5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.77, size = 17, normalized size = 0.57 \begin {gather*} e^{\left (e^{\left (e^{\left (x e^{\left (-5\right )} - 2 \, x + \frac {4}{x} - 3\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(e^(e^(x*e^(-5) - 2*x + 4/x - 3)))

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mupad [B]  time = 1.11, size = 20, normalized size = 0.67 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.64, size = 26, normalized size = 0.87 \begin {gather*} e^{e^{e^{\frac {x^{2} + \left (- 2 x^{2} - 3 x + 4\right ) e^{5}}{x e^{5}}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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