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

Optimal. Leaf size=25 \[ \frac {3-e^{6+e^{e^{-2 x} \left (3+e^4\right )}}}{x} \]

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Rubi [F]  time = 1.36, 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 {e^{-2 x} \left (-3 e^{2 x}+e^{6+e^{e^{-2 x} \left (3+e^4\right )}} \left (e^{2 x}+e^{e^{-2 x} \left (3+e^4\right )} \left (6 x+2 e^4 x\right )\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

3/x + Defer[Int][E^(6 + E^(E^(4 - 2*x) + 3/E^(2*x)))/x^2, x] + 2*(3 + E^4)*Defer[Int][E^(6 + E^(E^(4 - 2*x) +
3/E^(2*x)) + (1 + 3/E^4)*E^(4 - 2*x) - 2*x)/x, x]

Rubi steps

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

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Mathematica [A]  time = 0.66, size = 25, normalized size = 1.00 \begin {gather*} \frac {3-e^{6+e^{e^{-2 x} \left (3+e^4\right )}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3 - E^(6 + E^((3 + E^4)/E^(2*x))))/x

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fricas [A]  time = 0.63, size = 20, normalized size = 0.80 \begin {gather*} -\frac {e^{\left (e^{\left ({\left (e^{4} + 3\right )} e^{\left (-2 \, x\right )}\right )} + 6\right )} - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 25, normalized size = 1.00

method	result	size
risch	\(\frac {3}{x}-\frac {{\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{4}+3\right ) {\mathrm e}^{-2 x}}+6}}{x}\)	\(25\)
norman	\(\frac {\left (-{\mathrm e}^{2 x} {\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{4}+3\right ) {\mathrm e}^{-2 x}}+6}+3 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{x}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 28, normalized size = 1.12 \begin {gather*} -\frac {e^{\left (e^{\left (3 \, e^{\left (-2 \, x\right )} + e^{\left (-2 \, x + 4\right )}\right )} + 6\right )}}{x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.46, size = 27, normalized size = 1.08 \begin {gather*} -\frac {{\mathrm {e}}^{{\mathrm {e}}^{3\,{\mathrm {e}}^{-2\,x}}\,{\mathrm {e}}^{{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^4}}\,{\mathrm {e}}^6-3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(exp(exp(3*exp(-2*x))*exp(exp(-2*x)*exp(4)))*exp(6) - 3)/x

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sympy [A]  time = 0.27, size = 19, normalized size = 0.76 \begin {gather*} - \frac {e^{e^{\left (3 + e^{4}\right ) e^{- 2 x}} + 6}}{x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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