3.14.18 \(\int \frac {e^{-x} (e^x x^2+e^{\frac {1-e^{e^{e^{-x} (-x+e^x x^6)}}+x^2}{x}} (e^x (-1+x^2)+e^{e^{e^{-x} (-x+e^x x^6)}} (e^x+e^{e^{-x} (-x+e^x x^6)} (x-x^2-6 e^x x^6))))}{x^2} \, dx\)

Optimal. Leaf size=29 \[ e^{-\frac {-1+e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \]

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Rubi [F]  time = 20.82, 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^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.40, size = 30, normalized size = 1.03 \begin {gather*} e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^2 + E^((1 - E^E^((-x + E^x*x^6)/E^x) + x^2)/x)*(E^x*(-1 + x^2) + E^E^((-x + E^x*x^6)/E^x)*(E^
x + E^((-x + E^x*x^6)/E^x)*(x - x^2 - 6*E^x*x^6))))/(E^x*x^2),x]

[Out]

E^(x^(-1) - E^E^(-(x/E^x) + x^6)/x + x) + x

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fricas [A]  time = 0.70, size = 31, normalized size = 1.07 \begin {gather*} x + e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*
exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} e^{x} + {\left ({\left (x^{2} - 1\right )} e^{x} - {\left ({\left (6 \, x^{6} e^{x} + x^{2} - x\right )} e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )} - e^{x}\right )} e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )}\right )} e^{\left (-x\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*
exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.21, size = 31, normalized size = 1.07

method	result	size
risch	\(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{x \left (x^{5} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x)
)*exp((-exp(exp((x^6*exp(x)-x)/exp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.62, size = 26, normalized size = 0.90 \begin {gather*} x + e^{\left (x - \frac {e^{\left (e^{\left (x^{6} - x e^{\left (-x\right )}\right )}\right )}}{x} + \frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*
exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.09, size = 29, normalized size = 1.00 \begin {gather*} x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}}}{x}}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(x^2*exp(x) + exp((x^2 - exp(exp(-exp(-x)*(x - x^6*exp(x)))) + 1)/x)*(exp(exp(-exp(-x)*(x - x^6*e
xp(x))))*(exp(x) - exp(-exp(-x)*(x - x^6*exp(x)))*(6*x^6*exp(x) - x + x^2)) + exp(x)*(x^2 - 1))))/x^2,x)

[Out]

x + exp(-exp(exp(x^6)*exp(-x*exp(-x)))/x)*exp(1/x)*exp(x)

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sympy [A]  time = 15.26, size = 24, normalized size = 0.83 \begin {gather*} x + e^{\frac {x^{2} - e^{e^{\left (x^{6} e^{x} - x\right ) e^{- x}}} + 1}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-6*x**6*exp(x)-x**2+x)*exp((x**6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x**6*exp(x)-x)/exp(x)))+(x**
2-1)*exp(x))*exp((-exp(exp((x**6*exp(x)-x)/exp(x)))+x**2+1)/x)+exp(x)*x**2)/exp(x)/x**2,x)

[Out]

x + exp((x**2 - exp(exp((x**6*exp(x) - x)*exp(-x))) + 1)/x)

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