3.62.40 \(\int \frac {-4+e^{e^{\frac {e^x}{12}}+x} (-12 x^2-e^{\frac {e^x}{12}+x} x^2)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {4-12 e^{e^{\frac {e^x}{12}}+x} x}{x} \]

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Rubi [F]  time = 0.09, 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 {-4+e^{e^{\frac {e^x}{12}}+x} \left (-12 x^2-e^{\frac {e^x}{12}+x} x^2\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

4/x - 144*ExpIntegralEi[E^(E^x/12)] - Defer[Int][E^((12*E^(E^x/12) + E^x + 24*x)/12), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-12 e^{e^{\frac {e^x}{12}}+x}-e^{\frac {1}{12} \left (12 e^{\frac {e^x}{12}}+e^x+24 x\right )}-\frac {4}{x^2}\right ) \, dx\\ &=\frac {4}{x}-12 \int e^{e^{\frac {e^x}{12}}+x} \, dx-\int e^{\frac {1}{12} \left (12 e^{\frac {e^x}{12}}+e^x+24 x\right )} \, dx\\ &=\frac {4}{x}-12 \operatorname {Subst}\left (\int e^{e^{x/12}} \, dx,x,e^x\right )-\int e^{\frac {1}{12} \left (12 e^{\frac {e^x}{12}}+e^x+24 x\right )} \, dx\\ &=\frac {4}{x}-144 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^{\frac {e^x}{12}}\right )-\int e^{\frac {1}{12} \left (12 e^{\frac {e^x}{12}}+e^x+24 x\right )} \, dx\\ &=\frac {4}{x}-144 \text {Ei}\left (e^{\frac {e^x}{12}}\right )-\int e^{\frac {1}{12} \left (12 e^{\frac {e^x}{12}}+e^x+24 x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 21, normalized size = 0.95 \begin {gather*} -12 e^{e^{\frac {e^x}{12}}+x}+\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-12*E^(E^(E^x/12) + x) + 4/x

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fricas [A]  time = 0.50, size = 28, normalized size = 1.27 \begin {gather*} -\frac {4 \, {\left (3 \, x e^{\left ({\left (x e^{x} + e^{\left (x + \frac {1}{12} \, e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x)*exp(1/12*exp(x))-12*x^2)*exp(exp(1/12*exp(x))+x)-4)/x^2,x, algorithm="fricas")

[Out]

-4*(3*x*e^((x*e^x + e^(x + 1/12*e^x))*e^(-x)) - 1)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{2} e^{\left (x + \frac {1}{12} \, e^{x}\right )} + 12 \, x^{2}\right )} e^{\left (x + e^{\left (\frac {1}{12} \, e^{x}\right )}\right )} + 4}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x)*exp(1/12*exp(x))-12*x^2)*exp(exp(1/12*exp(x))+x)-4)/x^2,x, algorithm="giac")

[Out]

integrate(-((x^2*e^(x + 1/12*e^x) + 12*x^2)*e^(x + e^(1/12*e^x)) + 4)/x^2, x)

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maple [A]  time = 0.08, size = 17, normalized size = 0.77

method	result	size
risch	\(\frac {4}{x}-12 \,{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{x}}{12}}+x}\)	\(17\)
norman	\(\frac {4-12 x \,{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{x}}{12}}+x}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*exp(x)*exp(1/12*exp(x))-12*x^2)*exp(exp(1/12*exp(x))+x)-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

4/x-12*exp(exp(1/12*exp(x))+x)

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maxima [A]  time = 0.41, size = 16, normalized size = 0.73 \begin {gather*} \frac {4}{x} - 12 \, e^{\left (x + e^{\left (\frac {1}{12} \, e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(x)*exp(1/12*exp(x))-12*x^2)*exp(exp(1/12*exp(x))+x)-4)/x^2,x, algorithm="maxima")

[Out]

4/x - 12*e^(x + e^(1/12*e^x))

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mupad [B]  time = 0.10, size = 16, normalized size = 0.73 \begin {gather*} \frac {4}{x}-12\,{\mathrm {e}}^{{\left ({\mathrm {e}}^{{\mathrm {e}}^x}\right )}^{1/12}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + exp(exp(x)/12))*(12*x^2 + x^2*exp(exp(x)/12)*exp(x)) + 4)/x^2,x)

[Out]

4/x - 12*exp(exp(exp(x))^(1/12))*exp(x)

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sympy [A]  time = 0.22, size = 14, normalized size = 0.64 \begin {gather*} - 12 e^{x + e^{\frac {e^{x}}{12}}} + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2*exp(x)*exp(1/12*exp(x))-12*x**2)*exp(exp(1/12*exp(x))+x)-4)/x**2,x)

[Out]

-12*exp(x + exp(exp(x)/12)) + 4/x

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