3.20.72 \(\int \frac {e^{-4+\frac {e^4 (12-12 x)+4 x^2}{e^4 (-1+2 x)}} (-12 e^4-8 x+8 x^2)}{1-4 x+4 x^2} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {4 \left (3+x+x \left (-4+\frac {x}{e^4}\right )\right )}{-1+2 x}} \]

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Rubi [F]  time = 0.86, 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 (-4+\frac {e^4 (12-12 x)+4 x^2}{e^4 (-1+2 x)}\right ) \left (-12 e^4-8 x+8 x^2\right )}{1-4 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-4 + (E^4*(12 - 12*x) + 4*x^2)/(E^4*(-1 + 2*x)))*(-12*E^4 - 8*x + 8*x^2))/(1 - 4*x + 4*x^2),x]

[Out]

2*Defer[Int][E^((-4*(-4*E^4 + 5*E^4*x - x^2))/(E^4*(-1 + 2*x))), x] - 2*(1 + 6*E^4)*Defer[Int][1/(E^((4*(-4*E^
4 + 5*E^4*x - x^2))/(E^4*(-1 + 2*x)))*(-1 + 2*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-4+\frac {e^4 (12-12 x)+4 x^2}{e^4 (-1+2 x)}\right ) \left (-12 e^4-8 x+8 x^2\right )}{(-1+2 x)^2} \, dx\\ &=\int \frac {\exp \left (-\frac {4 \left (-4 e^4+5 e^4 x-x^2\right )}{e^4 (-1+2 x)}\right ) \left (-12 e^4-8 x+8 x^2\right )}{(1-2 x)^2} \, dx\\ &=\int \left (2 \exp \left (-\frac {4 \left (-4 e^4+5 e^4 x-x^2\right )}{e^4 (-1+2 x)}\right )-\frac {2 \exp \left (-\frac {4 \left (-4 e^4+5 e^4 x-x^2\right )}{e^4 (-1+2 x)}\right ) \left (1+6 e^4\right )}{(-1+2 x)^2}\right ) \, dx\\ &=2 \int \exp \left (-\frac {4 \left (-4 e^4+5 e^4 x-x^2\right )}{e^4 (-1+2 x)}\right ) \, dx-\left (2 \left (1+6 e^4\right )\right ) \int \frac {\exp \left (-\frac {4 \left (-4 e^4+5 e^4 x-x^2\right )}{e^4 (-1+2 x)}\right )}{(-1+2 x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 26, normalized size = 1.13 \begin {gather*} e^{\frac {4 \left (-3 e^4 (-1+x)+x^2\right )}{e^4 (-1+2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + (E^4*(12 - 12*x) + 4*x^2)/(E^4*(-1 + 2*x)))*(-12*E^4 - 8*x + 8*x^2))/(1 - 4*x + 4*x^2),x]

[Out]

E^((4*(-3*E^4*(-1 + x) + x^2))/(E^4*(-1 + 2*x)))

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fricas [A]  time = 1.05, size = 27, normalized size = 1.17 \begin {gather*} e^{\left (\frac {4 \, {\left (x^{2} - {\left (5 \, x - 4\right )} e^{4}\right )} e^{\left (-4\right )}}{2 \, x - 1} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(4)+8*x^2-8*x)*exp(((-12*x+12)*exp(4)+4*x^2)/(2*x-1)/exp(4))/(4*x^2-4*x+1)/exp(4),x, algorit
hm="fricas")

[Out]

e^(4*(x^2 - (5*x - 4)*e^4)*e^(-4)/(2*x - 1) + 4)

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giac [B]  time = 0.20, size = 53, normalized size = 2.30 \begin {gather*} e^{\left (\frac {4 \, x^{2}}{2 \, x e^{4} - e^{4}} - \frac {20 \, x e^{4}}{2 \, x e^{4} - e^{4}} + \frac {16 \, e^{4}}{2 \, x e^{4} - e^{4}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(4)+8*x^2-8*x)*exp(((-12*x+12)*exp(4)+4*x^2)/(2*x-1)/exp(4))/(4*x^2-4*x+1)/exp(4),x, algorit
hm="giac")

[Out]

e^(4*x^2/(2*x*e^4 - e^4) - 20*x*e^4/(2*x*e^4 - e^4) + 16*e^4/(2*x*e^4 - e^4) + 4)

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maple [A]  time = 0.38, size = 28, normalized size = 1.22

method	result	size
risch	\({\mathrm e}^{-\frac {4 \left (3 x \,{\mathrm e}^{4}-x^{2}-3 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{2 x -1}}\)	\(28\)
gosper	\({\mathrm e}^{-\frac {4 \left (3 x \,{\mathrm e}^{4}-x^{2}-3 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{2 x -1}}\)	\(30\)
norman	\(\frac {2 x \,{\mathrm e}^{\frac {\left (\left (-12 x +12\right ) {\mathrm e}^{4}+4 x^{2}\right ) {\mathrm e}^{-4}}{2 x -1}}-{\mathrm e}^{\frac {\left (\left (-12 x +12\right ) {\mathrm e}^{4}+4 x^{2}\right ) {\mathrm e}^{-4}}{2 x -1}}}{2 x -1}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*exp(4)+8*x^2-8*x)*exp(((-12*x+12)*exp(4)+4*x^2)/(2*x-1)/exp(4))/(4*x^2-4*x+1)/exp(4),x,method=_RETURN
VERBOSE)

[Out]

exp(-4*(3*x*exp(4)-x^2-3*exp(4))*exp(-4)/(2*x-1))

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maxima [A]  time = 0.78, size = 31, normalized size = 1.35 \begin {gather*} e^{\left (2 \, x e^{\left (-4\right )} + \frac {1}{2 \, x e^{4} - e^{4}} + \frac {6}{2 \, x - 1} + e^{\left (-4\right )} - 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(4)+8*x^2-8*x)*exp(((-12*x+12)*exp(4)+4*x^2)/(2*x-1)/exp(4))/(4*x^2-4*x+1)/exp(4),x, algorit
hm="maxima")

[Out]

e^(2*x*e^(-4) + 1/(2*x*e^4 - e^4) + 6/(2*x - 1) + e^(-4) - 6)

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mupad [B]  time = 1.33, size = 37, normalized size = 1.61 \begin {gather*} {\mathrm {e}}^{\frac {12}{2\,x-1}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{-4}}{2\,x-1}}\,{\mathrm {e}}^{-\frac {12\,x}{2\,x-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(-4)*(4*x^2 - exp(4)*(12*x - 12)))/(2*x - 1))*exp(-4)*(8*x + 12*exp(4) - 8*x^2))/(4*x^2 - 4*x +
1),x)

[Out]

exp(12/(2*x - 1))*exp((4*x^2*exp(-4))/(2*x - 1))*exp(-(12*x)/(2*x - 1))

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sympy [A]  time = 0.27, size = 22, normalized size = 0.96 \begin {gather*} e^{\frac {4 x^{2} + \left (12 - 12 x\right ) e^{4}}{\left (2 x - 1\right ) e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(4)+8*x**2-8*x)*exp(((-12*x+12)*exp(4)+4*x**2)/(2*x-1)/exp(4))/(4*x**2-4*x+1)/exp(4),x)

[Out]

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

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