3.93.3 \(\int \frac {e^{-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}} (-6+e^{1+x+x^2} (-3 x-6 x^2))}{1568 x^3} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}} \]

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Rubi [F]  time = 1.98, 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 (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right ) \left (-6+e^{1+x+x^2} \left (-3 x-6 x^2\right )\right )}{1568 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-1 - E^(1 + x + x^2) + (3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2))*(-6 + E^(1 + x + x^2)*(-3*x - 6*x^2)))
/(1568*x^3),x]

[Out]

(-3*Defer[Int][E^(-1 - E^(1 + x + x^2) + (3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2))/x^3, x])/784 - (3*Defer[Int]
[E^(-E^(1 + x + x^2) + (3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2) + x + x^2)/x^2, x])/1568 - (3*Defer[Int][E^(-E^
(1 + x + x^2) + (3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2) + x + x^2)/x, x])/784

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {\exp \left (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right ) \left (-6+e^{1+x+x^2} \left (-3 x-6 x^2\right )\right )}{x^3} \, dx}{1568}\\ &=\frac {\int \left (-\frac {6 \exp \left (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right )}{x^3}-\frac {3 \exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right ) (1+2 x)}{x^2}\right ) \, dx}{1568}\\ &=-\frac {3 \int \frac {\exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right ) (1+2 x)}{x^2} \, dx}{1568}-\frac {3}{784} \int \frac {\exp \left (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right )}{x^3} \, dx\\ &=-\frac {3 \int \left (\frac {\exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right )}{x^2}+\frac {2 \exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right )}{x}\right ) \, dx}{1568}-\frac {3}{784} \int \frac {\exp \left (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right )}{x^3} \, dx\\ &=-\frac {3 \int \frac {\exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right )}{x^2} \, dx}{1568}-\frac {3}{784} \int \frac {\exp \left (-1-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}\right )}{x^3} \, dx-\frac {3}{784} \int \frac {\exp \left (-e^{1+x+x^2}+\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}+x+x^2\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.64, size = 23, normalized size = 1.00 \begin {gather*} e^{\frac {3 e^{-1-e^{1+x+x^2}}}{1568 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 - E^(1 + x + x^2) + (3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2))*(-6 + E^(1 + x + x^2)*(-3*x - 6*
x^2)))/(1568*x^3),x]

[Out]

E^((3*E^(-1 - E^(1 + x + x^2)))/(1568*x^2))

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fricas [B]  time = 0.81, size = 47, normalized size = 2.04 \begin {gather*} e^{\left (-\frac {1568 \, x^{2} e^{\left (x^{2} + x + 1\right )} + 1568 \, x^{2} - 3 \, e^{\left (-e^{\left (x^{2} + x + 1\right )} - 1\right )}}{1568 \, x^{2}} + e^{\left (x^{2} + x + 1\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/1568*((-6*x^2-3*x)*exp(x^2+x+1)-6)*exp(3/1568/x^2/exp(exp(x^2+x+1)+1))/x^3/exp(exp(x^2+x+1)+1),x,
algorithm="fricas")

[Out]

e^(-1/1568*(1568*x^2*e^(x^2 + x + 1) + 1568*x^2 - 3*e^(-e^(x^2 + x + 1) - 1))/x^2 + e^(x^2 + x + 1) + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, {\left ({\left (2 \, x^{2} + x\right )} e^{\left (x^{2} + x + 1\right )} + 2\right )} e^{\left (\frac {3 \, e^{\left (-e^{\left (x^{2} + x + 1\right )} - 1\right )}}{1568 \, x^{2}} - e^{\left (x^{2} + x + 1\right )} - 1\right )}}{1568 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/1568*((-6*x^2-3*x)*exp(x^2+x+1)-6)*exp(3/1568/x^2/exp(exp(x^2+x+1)+1))/x^3/exp(exp(x^2+x+1)+1),x,
algorithm="giac")

[Out]

integrate(-3/1568*((2*x^2 + x)*e^(x^2 + x + 1) + 2)*e^(3/1568*e^(-e^(x^2 + x + 1) - 1)/x^2 - e^(x^2 + x + 1) -
1)/x^3, x)

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maple [A]  time = 0.09, size = 19, normalized size = 0.83

method	result	size
risch	\({\mathrm e}^{\frac {3 \,{\mathrm e}^{-{\mathrm e}^{x^{2}+x +1}-1}}{1568 x^{2}}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/1568*((-6*x^2-3*x)*exp(x^2+x+1)-6)*exp(3/1568/x^2/exp(exp(x^2+x+1)+1))/x^3/exp(exp(x^2+x+1)+1),x,method=
_RETURNVERBOSE)

[Out]

exp(3/1568/x^2*exp(-exp(x^2+x+1)-1))

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maxima [A]  time = 0.61, size = 18, normalized size = 0.78 \begin {gather*} e^{\left (\frac {3 \, e^{\left (-e^{\left (x^{2} + x + 1\right )} - 1\right )}}{1568 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/1568*((-6*x^2-3*x)*exp(x^2+x+1)-6)*exp(3/1568/x^2/exp(exp(x^2+x+1)+1))/x^3/exp(exp(x^2+x+1)+1),x,
algorithm="maxima")

[Out]

e^(3/1568*e^(-e^(x^2 + x + 1) - 1)/x^2)

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mupad [B]  time = 8.75, size = 19, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}\,\mathrm {e}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-1}}{1568\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- exp(x + x^2 + 1) - 1)*exp((3*exp(- exp(x + x^2 + 1) - 1))/(1568*x^2))*(exp(x + x^2 + 1)*(3*x + 6*x
^2) + 6))/(1568*x^3),x)

[Out]

exp((3*exp(-exp(x^2)*exp(1)*exp(x))*exp(-1))/(1568*x^2))

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sympy [A]  time = 1.00, size = 20, normalized size = 0.87 \begin {gather*} e^{\frac {3 e^{- e^{x^{2} + x + 1} - 1}}{1568 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/1568*((-6*x**2-3*x)*exp(x**2+x+1)-6)*exp(3/1568/x**2/exp(exp(x**2+x+1)+1))/x**3/exp(exp(x**2+x+1)+
1),x)

[Out]

exp(3*exp(-exp(x**2 + x + 1) - 1)/(1568*x**2))

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