3.92.20 \(\int \frac {e^{-x} (e^x x^2+e^{10 e^{-x}} (-e^{x+\frac {1+2 x}{x}}-10 e^{\frac {1+2 x}{x}} x^2))}{x^2} \, dx\)

Optimal. Leaf size=17 \[ -8+e^{2+10 e^{-x}+\frac {1}{x}}+x \]

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Rubi [F]  time = 0.30, 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^{10 e^{-x}} \left (-e^{x+\frac {1+2 x}{x}}-10 e^{\frac {1+2 x}{x}} x^2\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*x^2 + E^(10/E^x)*(-E^(x + (1 + 2*x)/x) - 10*E^((1 + 2*x)/x)*x^2))/(E^x*x^2),x]

[Out]

x - 10*Defer[Int][E^(2 + 10/E^x + x^(-1) - x), x] - Defer[Int][E^(2 + 10/E^x + x^(-1))/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-10 e^{2+10 e^{-x}+\frac {1}{x}-x}-\frac {e^{2+10 e^{-x}+\frac {1}{x}}}{x^2}\right ) \, dx\\ &=x-10 \int e^{2+10 e^{-x}+\frac {1}{x}-x} \, dx-\int \frac {e^{2+10 e^{-x}+\frac {1}{x}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 16, normalized size = 0.94 \begin {gather*} e^{2+10 e^{-x}+\frac {1}{x}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x^2 + E^(10/E^x)*(-E^(x + (1 + 2*x)/x) - 10*E^((1 + 2*x)/x)*x^2))/(E^x*x^2),x]

[Out]

E^(2 + 10/E^x + x^(-1)) + x

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fricas [A]  time = 0.60, size = 30, normalized size = 1.76 \begin {gather*} {\left (x e^{x} + e^{\left (\frac {x^{2} + 2 \, x + 1}{x} + 10 \, e^{\left (-x\right )}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp((2*x+1)/x)*exp(x)-10*x^2*exp((2*x+1)/x))*exp(5/exp(x))^2+exp(x)*x^2)/exp(x)/x^2,x, algorithm=
"fricas")

[Out]

(x*e^x + e^((x^2 + 2*x + 1)/x + 10*e^(-x)))*e^(-x)

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giac [A]  time = 0.18, size = 14, normalized size = 0.82 \begin {gather*} x + e^{\left (\frac {1}{x} + 10 \, e^{\left (-x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp((2*x+1)/x)*exp(x)-10*x^2*exp((2*x+1)/x))*exp(5/exp(x))^2+exp(x)*x^2)/exp(x)/x^2,x, algorithm=
"giac")

[Out]

x + e^(1/x + 10*e^(-x) + 2)

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maple [A]  time = 0.15, size = 20, normalized size = 1.18

method	result	size
risch	\({\mathrm e}^{\frac {10 x \,{\mathrm e}^{-x}+2 x +1}{x}}+x\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp((2*x+1)/x)*exp(x)-10*x^2*exp((2*x+1)/x))*exp(5/exp(x))^2+exp(x)*x^2)/exp(x)/x^2,x,method=_RETURNVER
BOSE)

[Out]

exp((10*x*exp(-x)+2*x+1)/x)+x

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maxima [A]  time = 0.45, size = 14, normalized size = 0.82 \begin {gather*} x + e^{\left (\frac {1}{x} + 10 \, e^{\left (-x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp((2*x+1)/x)*exp(x)-10*x^2*exp((2*x+1)/x))*exp(5/exp(x))^2+exp(x)*x^2)/exp(x)/x^2,x, algorithm=
"maxima")

[Out]

x + e^(1/x + 10*e^(-x) + 2)

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mupad [B]  time = 7.39, size = 16, normalized size = 0.94 \begin {gather*} x+{\mathrm {e}}^{10\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(x^2*exp(x) - exp(10*exp(-x))*(10*x^2*exp((2*x + 1)/x) + exp((2*x + 1)/x)*exp(x))))/x^2,x)

[Out]

x + exp(10*exp(-x))*exp(1/x)*exp(2)

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sympy [A]  time = 24.41, size = 15, normalized size = 0.88 \begin {gather*} x + e^{\frac {2 x + 1}{x}} e^{10 e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp((2*x+1)/x)*exp(x)-10*x**2*exp((2*x+1)/x))*exp(5/exp(x))**2+exp(x)*x**2)/exp(x)/x**2,x)

[Out]

x + exp((2*x + 1)/x)*exp(10*exp(-x))

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