3.55.16 \(\int \frac {e^{-2 x} (-e^{2 x} x^2+e^{\frac {2 e^{-2 x} (-e^4+e^{2 x} (-3+x))}{x}} (24 e^{2 x}+e^4 (8+16 x)))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ 4 \left (5+e^{\frac {2 \left (-3-e^{4-2 x}+x\right )}{x}}\right )-x \]

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Rubi [F]  time = 1.67, 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 (-e^{2 x} x^2+\exp \left (\frac {2 e^{-2 x} \left (-e^4+e^{2 x} (-3+x)\right )}{x}\right ) \left (24 e^{2 x}+e^4 (8+16 x)\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-(E^(2*x)*x^2) + E^((2*(-E^4 + E^(2*x)*(-3 + x)))/(E^(2*x)*x))*(24*E^(2*x) + E^4*(8 + 16*x)))/(E^(2*x)*x^
2),x]

[Out]

-x + 24*Defer[Int][E^((2*(-3 - E^(4 - 2*x) + x))/x)/x^2, x] + 8*Defer[Int][E^(4 - (2*(3 + E^(4 - 2*x) - x + x^
2))/x)/x^2, x] + 16*Defer[Int][E^(4 - (2*(3 + E^(4 - 2*x) - x + x^2))/x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {8 e^{-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}} \left (3 e^{2 x}+e^4 (1+2 x)\right )}{x^2}\right ) \, dx\\ &=-x+8 \int \frac {e^{-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}} \left (3 e^{2 x}+e^4 (1+2 x)\right )}{x^2} \, dx\\ &=-x+8 \int \left (\frac {3 e^{2 x-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x^2}+\frac {e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}} (1+2 x)}{x^2}\right ) \, dx\\ &=-x+8 \int \frac {e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}} (1+2 x)}{x^2} \, dx+24 \int \frac {e^{2 x-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x^2} \, dx\\ &=-x+8 \int \left (\frac {e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x^2}+\frac {2 e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x}\right ) \, dx+24 \int \frac {e^{\frac {2 \left (-3-e^{4-2 x}+x\right )}{x}}}{x^2} \, dx\\ &=-x+8 \int \frac {e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x^2} \, dx+16 \int \frac {e^{4-\frac {2 \left (3+e^{4-2 x}-x+x^2\right )}{x}}}{x} \, dx+24 \int \frac {e^{\frac {2 \left (-3-e^{4-2 x}+x\right )}{x}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 27, normalized size = 1.00 \begin {gather*} 4 e^{2-\frac {6}{x}-\frac {2 e^{4-2 x}}{x}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(E^(2*x)*x^2) + E^((2*(-E^4 + E^(2*x)*(-3 + x)))/(E^(2*x)*x))*(24*E^(2*x) + E^4*(8 + 16*x)))/(E^(2
*x)*x^2),x]

[Out]

4*E^(2 - 6/x - (2*E^(4 - 2*x))/x) - x

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fricas [A]  time = 0.66, size = 29, normalized size = 1.07 \begin {gather*} -x + 4 \, e^{\left (\frac {2 \, {\left ({\left (x - 3\right )} e^{\left (2 \, x\right )} - e^{4}\right )} e^{\left (-2 \, x\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*exp(2*x)+(16*x+8)*exp(4))*exp(((x-3)*exp(2*x)-exp(4))/x/exp(2*x))^2-exp(2*x)*x^2)/exp(2*x)/x^2,
x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.27, size = 23, normalized size = 0.85 \begin {gather*} -x + 4 \, e^{\left (\frac {2 \, {\left (x - e^{\left (-2 \, x + 4\right )} - 3\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*exp(2*x)+(16*x+8)*exp(4))*exp(((x-3)*exp(2*x)-exp(4))/x/exp(2*x))^2-exp(2*x)*x^2)/exp(2*x)/x^2,
x, algorithm="giac")

[Out]

-x + 4*e^(2*(x - e^(-2*x + 4) - 3)/x)

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maple [A]  time = 0.22, size = 33, normalized size = 1.22

method	result	size
risch	\(4 \,{\mathrm e}^{-\frac {2 \left (-x \,{\mathrm e}^{2 x}+{\mathrm e}^{4}+3 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{x}}-x\)	\(33\)
norman	\(\frac {\left (-{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{2 x} {\mathrm e}^{\frac {2 \left (\left (x -3\right ) {\mathrm e}^{2 x}-{\mathrm e}^{4}\right ) {\mathrm e}^{-2 x}}{x}}\right ) {\mathrm e}^{-2 x}}{x}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*exp(2*x)+(16*x+8)*exp(4))*exp(((x-3)*exp(2*x)-exp(4))/x/exp(2*x))^2-exp(2*x)*x^2)/exp(2*x)/x^2,x,meth
od=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.61, size = 25, normalized size = 0.93 \begin {gather*} -x + 4 \, e^{\left (-\frac {2 \, e^{\left (-2 \, x + 4\right )}}{x} - \frac {6}{x} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*exp(2*x)+(16*x+8)*exp(4))*exp(((x-3)*exp(2*x)-exp(4))/x/exp(2*x))^2-exp(2*x)*x^2)/exp(2*x)/x^2,
x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.64, size = 26, normalized size = 0.96 \begin {gather*} 4\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^4}{x}}\,{\mathrm {e}}^2\,{\mathrm {e}}^{-\frac {6}{x}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2*x)*(exp(-(2*exp(-2*x)*(exp(4) - exp(2*x)*(x - 3)))/x)*(24*exp(2*x) + exp(4)*(16*x + 8)) - x^2*exp(
2*x)))/x^2,x)

[Out]

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

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sympy [A]  time = 0.26, size = 26, normalized size = 0.96 \begin {gather*} - x + 4 e^{\frac {2 \left (\left (x - 3\right ) e^{2 x} - e^{4}\right ) e^{- 2 x}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*exp(2*x)+(16*x+8)*exp(4))*exp(((x-3)*exp(2*x)-exp(4))/x/exp(2*x))**2-exp(2*x)*x**2)/exp(2*x)/x*
*2,x)

[Out]

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

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