3.96.96 \(\int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} (1+\frac {e^{25+e^x+x} (-4+4 x-4 x^2+e^x (4 x-4 x^2))}{x}) \, dx\)

Optimal. Leaf size=22 \[ e^{e^{-\frac {4 e^{25+e^x+x} (-1+x)}{x}} x} \]

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Rubi [F]  time = 15.34, 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 \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x)*(1 + (E^(25 + E^x + x)*(-4 + 4
*x - 4*x^2 + E^x*(4*x - 4*x^2)))/x),x]

[Out]

Defer[Int][E^(x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x), x] + 4*Defer[Int][E^(2
5 + E^x + x + x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x), x] + 4*Defer[Int][E^(2
5 + E^x + 2*x + x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x), x] - 4*Defer[Int][E^
(25 + E^x + x + x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x)/x, x] - 4*Defer[Int][
E^(25 + E^x + x + x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x)*x, x] - 4*Defer[Int
][E^(25 + E^x + 2*x + x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right )-\frac {4 \exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \left (1-x-e^x x+x^2+e^x x^2\right )}{x}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \left (1-x-e^x x+x^2+e^x x^2\right )}{x} \, dx\right )+\int \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx\\ &=-\left (4 \int \left (\exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) (-1+x)+\frac {\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \left (1-x+x^2\right )}{x}\right ) \, dx\right )+\int \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx\\ &=-\left (4 \int \exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) (-1+x) \, dx\right )-4 \int \frac {\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \left (1-x+x^2\right )}{x} \, dx+\int \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx\\ &=-\left (4 \int \left (-\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right )+\frac {\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right )}{x}+\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) x\right ) \, dx\right )-4 \int \left (-\exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right )+\exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) x\right ) \, dx+\int \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx\\ &=4 \int \exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx+4 \int \exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx-4 \int \frac {\exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right )}{x} \, dx-4 \int \exp \left (25+e^x+x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) x \, dx-4 \int \exp \left (25+e^x+2 x+e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) x \, dx+\int \exp \left (e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.24, size = 22, normalized size = 1.00 \begin {gather*} e^{e^{-\frac {4 e^{25+e^x+x} (-1+x)}{x}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x))/x)*(1 + (E^(25 + E^x + x)*(
-4 + 4*x - 4*x^2 + E^x*(4*x - 4*x^2)))/x),x]

[Out]

E^(x/E^((4*E^(25 + E^x + x)*(-1 + x))/x))

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fricas [A]  time = 0.52, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (x e^{\left (-4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x*e^(-4*(x - 1)*e^(x + e^x + log(e^25/x))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -{\left (4 \, {\left (x^{2} + {\left (x^{2} - x\right )} e^{x} - x + 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )} - 1\right )} e^{\left (x e^{\left (-4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )} - 4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(4*(x^2 + (x^2 - x)*e^x - x + 1)*e^(x + e^x + log(e^25/x)) - 1)*e^(x*e^(-4*(x - 1)*e^(x + e^x + log
(e^25/x))) - 4*(x - 1)*e^(x + e^x + log(e^25/x))), x)

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maple [A]  time = 0.40, size = 19, normalized size = 0.86

method	result	size
risch	\({\mathrm e}^{x \,{\mathrm e}^{-\frac {4 \left (x -1\right ) {\mathrm e}^{25+{\mathrm e}^{x}+x}}{x}}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(ln(exp(25)/x)+exp(x)+x)+1)*exp(x/exp((4*x-4)*exp(ln(exp(25)/x)+exp(
x)+x)))/exp((4*x-4)*exp(ln(exp(25)/x)+exp(x)+x)),x,method=_RETURNVERBOSE)

[Out]

exp(x*exp(-4*(x-1)*exp(25+exp(x)+x)/x))

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maxima [A]  time = 0.72, size = 24, normalized size = 1.09 \begin {gather*} e^{\left (x e^{\left (\frac {4 \, e^{\left (x + e^{x} + 25\right )}}{x} - 4 \, e^{\left (x + e^{x} + 25\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x*e^(4*e^(x + e^x + 25)/x - 4*e^(x + e^x + 25)))

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mupad [B]  time = 8.11, size = 26, normalized size = 1.18 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(x + log(exp(25)/x) + exp(x))*(4*x - 4))*exp(x*exp(-exp(x + log(exp(25)/x) + exp(x))*(4*x - 4)))*(
exp(x + log(exp(25)/x) + exp(x))*(4*x + exp(x)*(4*x - 4*x^2) - 4*x^2 - 4) + 1),x)

[Out]

exp(x*exp((4*exp(exp(x))*exp(25)*exp(x))/x)*exp(-4*exp(exp(x))*exp(25)*exp(x)))

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sympy [A]  time = 39.98, size = 20, normalized size = 0.91 \begin {gather*} e^{x e^{- \frac {\left (4 x - 4\right ) e^{25} e^{x + e^{x}}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*exp(-(4*x - 4)*exp(25)*exp(x + exp(x))/x))

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