3.85.75 \(\int \frac {e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} (-25-e^8+2 x)}{1+e^8} \, dx\)

Optimal. Leaf size=20 \[ e^{e^{-\frac {2 x}{1+e^8}} (12-x)} \]

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Rubi [F]  time = 0.48, 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^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right )}{1+e^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((12 - x)/E^((2*x)/(1 + E^8)) - (2*x)/(1 + E^8))*(-25 - E^8 + 2*x))/(1 + E^8),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (-25-e^8+2 x\right ) \, dx}{1+e^8}\\ &=\frac {\int \left (-25 e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \left (1+\frac {e^8}{25}\right )+2 e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} x\right ) \, dx}{1+e^8}\\ &=\frac {2 \int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} x \, dx}{1+e^8}-\frac {\left (25+e^8\right ) \int e^{e^{-\frac {2 x}{1+e^8}} (12-x)-\frac {2 x}{1+e^8}} \, dx}{1+e^8}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 35, normalized size = 1.75 \begin {gather*} -\frac {e^{-e^{-\frac {2 x}{1+e^8}} (-12+x)} \left (-1-e^8\right )}{1+e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((12 - x)/E^((2*x)/(1 + E^8)) - (2*x)/(1 + E^8))*(-25 - E^8 + 2*x))/(1 + E^8),x]

[Out]

-((-1 - E^8)/(E^((-12 + x)/E^((2*x)/(1 + E^8)))*(1 + E^8)))

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fricas [B]  time = 0.64, size = 43, normalized size = 2.15 \begin {gather*} e^{\left (-\frac {{\left ({\left (x - 12\right )} e^{8} + x - 12\right )} e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )} + 2 \, x}{e^{8} + 1} + \frac {2 \, x}{e^{8} + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp(2*x/(exp(2)^4+1)),x, algorithm
="fricas")

[Out]

e^(-(((x - 12)*e^8 + x - 12)*e^(-2*x/(e^8 + 1)) + 2*x)/(e^8 + 1) + 2*x/(e^8 + 1))

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giac [B]  time = 0.54, size = 212, normalized size = 10.60 \begin {gather*} \frac {{\left (e^{\left (-\frac {2 \, x e^{8} + x e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1} + 8\right )} + x e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1}\right )} - 12 \, e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1} + 8\right )} - 12 \, e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1}\right )}}{e^{16} + e^{8}} + 16\right )} + e^{\left (-\frac {2 \, x e^{8} + x e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1} + 8\right )} + x e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1}\right )} - 12 \, e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1} + 8\right )} - 12 \, e^{\left (-\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1}\right )}}{e^{16} + e^{8}} + 8\right )}\right )} e^{\left (\frac {2 \, {\left (x - 4 \, e^{8} - 4\right )}}{e^{8} + 1}\right )}}{e^{8} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp(2*x/(exp(2)^4+1)),x, algorithm
="giac")

[Out]

(e^(-(2*x*e^8 + x*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1) + 8) + x*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1)) - 12*e^(-2*(x - 4*
e^8 - 4)/(e^8 + 1) + 8) - 12*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1)))/(e^16 + e^8) + 16) + e^(-(2*x*e^8 + x*e^(-2*(x
- 4*e^8 - 4)/(e^8 + 1) + 8) + x*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1)) - 12*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1) + 8) - 1
2*e^(-2*(x - 4*e^8 - 4)/(e^8 + 1)))/(e^16 + e^8) + 8))*e^(2*(x - 4*e^8 - 4)/(e^8 + 1))/(e^8 + 1)

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maple [A]  time = 0.10, size = 17, normalized size = 0.85

method	result	size
risch	\({\mathrm e}^{-\left (x -12\right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}\)	\(17\)
norman	\({\mathrm e}^{\left (12-x \right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{8}+1}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp(2*x/(exp(2)^4+1)),x,method=_RETURNVE
RBOSE)

[Out]

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

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maxima [A]  time = 0.65, size = 27, normalized size = 1.35 \begin {gather*} e^{\left (-x e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )} + 12 \, e^{\left (-\frac {2 \, x}{e^{8} + 1}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)^4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)^4+1)))/(exp(2)^4+1)/exp(2*x/(exp(2)^4+1)),x, algorithm
="maxima")

[Out]

e^(-x*e^(-2*x/(e^8 + 1)) + 12*e^(-2*x/(e^8 + 1)))

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mupad [B]  time = 0.23, size = 28, normalized size = 1.40 \begin {gather*} {\mathrm {e}}^{12\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^8+1}}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^8+1}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*x)/(exp(8) + 1))*exp(-exp(-(2*x)/(exp(8) + 1))*(x - 12))*(exp(8) - 2*x + 25))/(exp(8) + 1),x)

[Out]

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

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sympy [A]  time = 0.26, size = 14, normalized size = 0.70 \begin {gather*} e^{\left (12 - x\right ) e^{- \frac {2 x}{1 + e^{8}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(2)**4+2*x-25)*exp((12-x)/exp(2*x/(exp(2)**4+1)))/(exp(2)**4+1)/exp(2*x/(exp(2)**4+1)),x)

[Out]

exp((12 - x)*exp(-2*x/(1 + exp(8))))

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