3.81.99 \(\int \frac {3 x+\frac {e^{-e^x+e^{x-x^2}-5 x} (-3-15 x-3 e^x x+e^{x-x^2} (3 x-6 x^2))}{x}}{x} \, dx\)

Optimal. Leaf size=28 \[ 3 \left (\frac {e^{-e^x+e^{x-x^2}-5 x}}{x}+x\right ) \]

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Rubi [F]  time = 3.60, 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 {3 x+\frac {e^{-e^x+e^{x-x^2}-5 x} \left (-3-15 x-3 e^x x+e^{x-x^2} \left (3 x-6 x^2\right )\right )}{x}}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*x + (E^(-E^x + E^(x - x^2) - 5*x)*(-3 - 15*x - 3*E^x*x + E^(x - x^2)*(3*x - 6*x^2)))/x)/x,x]

[Out]

3*x - 6*Defer[Int][E^(-E^x + E^(x - x^2) - x*(4 + x)), x] - 3*Defer[Int][E^(-E^x + E^((1 - x)*x) - 5*x)/x^2, x
] - 15*Defer[Int][E^(-E^x + E^((1 - x)*x) - 5*x)/x, x] - 3*Defer[Int][E^(-E^x + E^(x - x^2) - 4*x)/x, x] + 3*D
efer[Int][E^(-E^x + E^(x - x^2) - x*(4 + x))/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {3 e^{-e^x+e^{x-x^2}-x (4+x)} (-1+2 x)}{x}+\frac {3 e^{-e^x-5 x} \left (-e^{e^{x-x^2}}-5 e^{e^{x-x^2}} x-e^{e^{x-x^2}+x} x+e^{e^x+5 x} x^2\right )}{x^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{-e^x+e^{x-x^2}-x (4+x)} (-1+2 x)}{x} \, dx\right )+3 \int \frac {e^{-e^x-5 x} \left (-e^{e^{x-x^2}}-5 e^{e^{x-x^2}} x-e^{e^{x-x^2}+x} x+e^{e^x+5 x} x^2\right )}{x^2} \, dx\\ &=3 \int \left (1+\frac {e^{-e^x+e^{(1-x) x}-5 x} (-1-5 x)}{x^2}-\frac {e^{-e^x+e^{x-x^2}-4 x}}{x}\right ) \, dx-3 \int \left (2 e^{-e^x+e^{x-x^2}-x (4+x)}-\frac {e^{-e^x+e^{x-x^2}-x (4+x)}}{x}\right ) \, dx\\ &=3 x+3 \int \frac {e^{-e^x+e^{(1-x) x}-5 x} (-1-5 x)}{x^2} \, dx-3 \int \frac {e^{-e^x+e^{x-x^2}-4 x}}{x} \, dx+3 \int \frac {e^{-e^x+e^{x-x^2}-x (4+x)}}{x} \, dx-6 \int e^{-e^x+e^{x-x^2}-x (4+x)} \, dx\\ &=3 x+3 \int \left (-\frac {e^{-e^x+e^{(1-x) x}-5 x}}{x^2}-\frac {5 e^{-e^x+e^{(1-x) x}-5 x}}{x}\right ) \, dx-3 \int \frac {e^{-e^x+e^{x-x^2}-4 x}}{x} \, dx+3 \int \frac {e^{-e^x+e^{x-x^2}-x (4+x)}}{x} \, dx-6 \int e^{-e^x+e^{x-x^2}-x (4+x)} \, dx\\ &=3 x-3 \int \frac {e^{-e^x+e^{(1-x) x}-5 x}}{x^2} \, dx-3 \int \frac {e^{-e^x+e^{x-x^2}-4 x}}{x} \, dx+3 \int \frac {e^{-e^x+e^{x-x^2}-x (4+x)}}{x} \, dx-6 \int e^{-e^x+e^{x-x^2}-x (4+x)} \, dx-15 \int \frac {e^{-e^x+e^{(1-x) x}-5 x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.89, size = 29, normalized size = 1.04 \begin {gather*} \frac {3 e^{-e^x+e^{x-x^2}-5 x}}{x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*x + (E^(-E^x + E^(x - x^2) - 5*x)*(-3 - 15*x - 3*E^x*x + E^(x - x^2)*(3*x - 6*x^2)))/x)/x,x]

[Out]

(3*E^(-E^x + E^(x - x^2) - 5*x))/x + 3*x

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fricas [A]  time = 0.81, size = 27, normalized size = 0.96 \begin {gather*} 3 \, x + 3 \, e^{\left (-5 \, x + e^{\left (-x^{2} + x\right )} - e^{x} - \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x+(-6*x^2+3*x)*exp(-x^2+x)-15*x-3)*exp(-log(x)-exp(x)+exp(-x^2+x)-5*x)+3*x)/x,x, algorit
hm="fricas")

[Out]

3*x + 3*e^(-5*x + e^(-x^2 + x) - e^x - log(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, {\left ({\left ({\left (2 \, x^{2} - x\right )} e^{\left (-x^{2} + x\right )} + x e^{x} + 5 \, x + 1\right )} e^{\left (-5 \, x + e^{\left (-x^{2} + x\right )} - e^{x} - \log \relax (x)\right )} - x\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x+(-6*x^2+3*x)*exp(-x^2+x)-15*x-3)*exp(-log(x)-exp(x)+exp(-x^2+x)-5*x)+3*x)/x,x, algorit
hm="giac")

[Out]

integrate(-3*(((2*x^2 - x)*e^(-x^2 + x) + x*e^x + 5*x + 1)*e^(-5*x + e^(-x^2 + x) - e^x - log(x)) - x)/x, x)

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maple [A]  time = 0.10, size = 26, normalized size = 0.93

method	result	size
risch	\(3 x +\frac {3 \,{\mathrm e}^{-{\mathrm e}^{x}+{\mathrm e}^{-x \left (x -1\right )}-5 x}}{x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*exp(x)*x+(-6*x^2+3*x)*exp(-x^2+x)-15*x-3)*exp(-ln(x)-exp(x)+exp(-x^2+x)-5*x)+3*x)/x,x,method=_RETURNV
ERBOSE)

[Out]

3*x+3/x*exp(-exp(x)+exp(-x*(x-1))-5*x)

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maxima [A]  time = 0.51, size = 26, normalized size = 0.93 \begin {gather*} 3 \, x + \frac {3 \, e^{\left (-5 \, x + e^{\left (-x^{2} + x\right )} - e^{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x+(-6*x^2+3*x)*exp(-x^2+x)-15*x-3)*exp(-log(x)-exp(x)+exp(-x^2+x)-5*x)+3*x)/x,x, algorit
hm="maxima")

[Out]

3*x + 3*e^(-5*x + e^(-x^2 + x) - e^x)/x

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mupad [B]  time = 5.64, size = 28, normalized size = 1.00 \begin {gather*} 3\,x+\frac {3\,{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-{\mathrm {e}}^x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - exp(exp(x - x^2) - 5*x - exp(x) - log(x))*(15*x - exp(x - x^2)*(3*x - 6*x^2) + 3*x*exp(x) + 3))/x,x
)

[Out]

3*x + (3*exp(-5*x)*exp(exp(-x^2)*exp(x))*exp(-exp(x)))/x

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sympy [A]  time = 0.25, size = 20, normalized size = 0.71 \begin {gather*} 3 x + \frac {3 e^{- 5 x - e^{x} + e^{- x^{2} + x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x+(-6*x**2+3*x)*exp(-x**2+x)-15*x-3)*exp(-ln(x)-exp(x)+exp(-x**2+x)-5*x)+3*x)/x,x)

[Out]

3*x + 3*exp(-5*x - exp(x) + exp(-x**2 + x))/x

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