3.4.83 \(\int \frac {e^{e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}} (-2+2 e^5-7 x)}{-32+48 x-24 x^2+4 x^3} \, dx\) [383]

Optimal. Leaf size=22 \[ e^{e^{\frac {-6-e^5+7 x}{(-4+2 x)^2}}} \]

[Out]

exp(exp((-exp(5)+7*x-6)/(2*x-4)^2))

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Rubi [F]
time = 2.04, 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 (e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}\right ) \left (-2+2 e^5-7 x\right )}{-32+48 x-24 x^2+4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2)) + (-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2))*(-2 + 2*E^5 - 7*x))/(
-32 + 48*x - 24*x^2 + 4*x^3),x]

[Out]

-1/2*((8 - E^5)*Defer[Int][E^(E^((-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2)) + (-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2))
/(-2 + x)^3, x]) - (7*Defer[Int][E^(E^((-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2)) + (-6 - E^5 + 7*x)/(16 - 16*x + 4
*x^2))/(-2 + x)^2, x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\exp \left (e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}\right ) \left (-8+e^5\right )}{2 (-2+x)^3}-\frac {7 \exp \left (e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}\right )}{4 (-2+x)^2}\right ) \, dx\\ &=-\left (\frac {7}{4} \int \frac {\exp \left (e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}\right )}{(-2+x)^2} \, dx\right )+\frac {1}{2} \left (-8+e^5\right ) \int \frac {\exp \left (e^{\frac {-6-e^5+7 x}{16-16 x+4 x^2}}+\frac {-6-e^5+7 x}{16-16 x+4 x^2}\right )}{(-2+x)^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.77, size = 21, normalized size = 0.95 \begin {gather*} e^{e^{-\frac {6+e^5-7 x}{4 (-2+x)^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2)) + (-6 - E^5 + 7*x)/(16 - 16*x + 4*x^2))*(-2 + 2*E^5 - 7
*x))/(-32 + 48*x - 24*x^2 + 4*x^3),x]

[Out]

E^E^(-1/4*(6 + E^5 - 7*x)/(-2 + x)^2)

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Maple [A]
time = 0.24, size = 17, normalized size = 0.77

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{-\frac {{\mathrm e}^{5}-7 x +6}{4 \left (x -2\right )^{2}}}}\)	\(17\)
norman	\(\frac {x^{2} {\mathrm e}^{{\mathrm e}^{\frac {-{\mathrm e}^{5}+7 x -6}{4 x^{2}-16 x +16}}}-4 x \,{\mathrm e}^{{\mathrm e}^{\frac {-{\mathrm e}^{5}+7 x -6}{4 x^{2}-16 x +16}}}+4 \,{\mathrm e}^{{\mathrm e}^{\frac {-{\mathrm e}^{5}+7 x -6}{4 x^{2}-16 x +16}}}}{\left (x -2\right )^{2}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(5)-7*x-2)*exp((-exp(5)+7*x-6)/(4*x^2-16*x+16))*exp(exp((-exp(5)+7*x-6)/(4*x^2-16*x+16)))/(4*x^3-24*
x^2+48*x-32),x,method=_RETURNVERBOSE)

[Out]

exp(exp(-1/4*(exp(5)-7*x+6)/(x-2)^2))

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Maxima [A]
time = 0.98, size = 36, normalized size = 1.64 \begin {gather*} e^{\left (e^{\left (-\frac {e^{5}}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {2}{x^{2} - 4 \, x + 4} + \frac {7}{4 \, {\left (x - 2\right )}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)-7*x-2)*exp((-exp(5)+7*x-6)/(4*x^2-16*x+16))*exp(exp((-exp(5)+7*x-6)/(4*x^2-16*x+16)))/(4*x
^3-24*x^2+48*x-32),x, algorithm="maxima")

[Out]

e^(e^(-1/4*e^5/(x^2 - 4*x + 4) + 2/(x^2 - 4*x + 4) + 7/4/(x - 2)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (18) = 36\).
time = 0.31, size = 76, normalized size = 3.45 \begin {gather*} e^{\left (\frac {4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (\frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}}\right )} + 7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)-7*x-2)*exp((-exp(5)+7*x-6)/(4*x^2-16*x+16))*exp(exp((-exp(5)+7*x-6)/(4*x^2-16*x+16)))/(4*x
^3-24*x^2+48*x-32),x, algorithm="fricas")

[Out]

e^(1/4*(4*(x^2 - 4*x + 4)*e^(1/4*(7*x - e^5 - 6)/(x^2 - 4*x + 4)) + 7*x - e^5 - 6)/(x^2 - 4*x + 4) - 1/4*(7*x
- e^5 - 6)/(x^2 - 4*x + 4))

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Sympy [A]
time = 0.23, size = 20, normalized size = 0.91 \begin {gather*} e^{e^{\frac {7 x - e^{5} - 6}{4 x^{2} - 16 x + 16}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)-7*x-2)*exp((-exp(5)+7*x-6)/(4*x**2-16*x+16))*exp(exp((-exp(5)+7*x-6)/(4*x**2-16*x+16)))/(4
*x**3-24*x**2+48*x-32),x)

[Out]

exp(exp((7*x - exp(5) - 6)/(4*x**2 - 16*x + 16)))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (18) = 36\).
time = 1.86, size = 136, normalized size = 6.18 \begin {gather*} e^{\left (\frac {x^{2} e^{5} + 16 \, x^{2} e^{\left (\frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}}\right )} + 6 \, x^{2} - 4 \, x e^{5} - 64 \, x e^{\left (\frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}}\right )} + 4 \, x + 64 \, e^{\left (\frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}}\right )}}{16 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {7 \, x - e^{5} - 6}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {1}{16} \, e^{5} - \frac {3}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(5)-7*x-2)*exp((-exp(5)+7*x-6)/(4*x^2-16*x+16))*exp(exp((-exp(5)+7*x-6)/(4*x^2-16*x+16)))/(4*x
^3-24*x^2+48*x-32),x, algorithm="giac")

[Out]

e^(1/16*(x^2*e^5 + 16*x^2*e^(1/4*(7*x - e^5 - 6)/(x^2 - 4*x + 4)) + 6*x^2 - 4*x*e^5 - 64*x*e^(1/4*(7*x - e^5 -
6)/(x^2 - 4*x + 4)) + 4*x + 64*e^(1/4*(7*x - e^5 - 6)/(x^2 - 4*x + 4)))/(x^2 - 4*x + 4) - 1/4*(7*x - e^5 - 6)
/(x^2 - 4*x + 4) - 1/16*e^5 - 3/8)

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Mupad [B]
time = 1.49, size = 50, normalized size = 2.27 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {7\,x}{4\,x^2-16\,x+16}}\,{\mathrm {e}}^{-\frac {3}{2\,x^2-8\,x+8}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^5}{4\,x^2-16\,x+16}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(exp(5) - 7*x + 6)/(4*x^2 - 16*x + 16))*exp(exp(-(exp(5) - 7*x + 6)/(4*x^2 - 16*x + 16)))*(7*x - 2*
exp(5) + 2))/(48*x - 24*x^2 + 4*x^3 - 32),x)

[Out]

exp(exp((7*x)/(4*x^2 - 16*x + 16))*exp(-3/(2*x^2 - 8*x + 8))*exp(-exp(5)/(4*x^2 - 16*x + 16)))

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