3.76.60 \(\int \frac {e^{-x} (e^{10 e^{-x}} (-2 e^x-10 x)-e^{e^x+2 x} x^3+e^{5 e^{-x}} (4 e^x+10 x)+e^x (-6-x^3))}{x^3} \, dx\)

Optimal. Leaf size=33 \[ 2-e^{e^x}+\frac {2+\left (1-e^{5 e^{-x}}\right )^2}{x^2}-x \]

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Rubi [F]  time = 0.97, 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^{10 e^{-x}} \left (-2 e^x-10 x\right )-e^{e^x+2 x} x^3+e^{5 e^{-x}} \left (4 e^x+10 x\right )+e^x \left (-6-x^3\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(10/E^x)*(-2*E^x - 10*x) - E^(E^x + 2*x)*x^3 + E^(5/E^x)*(4*E^x + 10*x) + E^x*(-6 - x^3))/(E^x*x^3),x]

[Out]

-E^E^x + 3/x^2 - x + 4*Defer[Int][E^(5/E^x)/x^3, x] - 2*Defer[Int][E^(10/E^x)/x^3, x] + 10*Defer[Int][E^(5/E^x
- x)/x^2, x] - 10*Defer[Int][E^(10/E^x - x)/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{e^x+x}-\frac {10 e^{5 e^{-x}-x} \left (-1+e^{5 e^{-x}}\right )}{x^2}-\frac {6-4 e^{5 e^{-x}}+2 e^{10 e^{-x}}+x^3}{x^3}\right ) \, dx\\ &=-\left (10 \int \frac {e^{5 e^{-x}-x} \left (-1+e^{5 e^{-x}}\right )}{x^2} \, dx\right )-\int e^{e^x+x} \, dx-\int \frac {6-4 e^{5 e^{-x}}+2 e^{10 e^{-x}}+x^3}{x^3} \, dx\\ &=-\left (10 \int \left (-\frac {e^{5 e^{-x}-x}}{x^2}+\frac {e^{10 e^{-x}-x}}{x^2}\right ) \, dx\right )-\int \left (-\frac {4 e^{5 e^{-x}}}{x^3}+\frac {2 e^{10 e^{-x}}}{x^3}+\frac {6+x^3}{x^3}\right ) \, dx-\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=-e^{e^x}-2 \int \frac {e^{10 e^{-x}}}{x^3} \, dx+4 \int \frac {e^{5 e^{-x}}}{x^3} \, dx+10 \int \frac {e^{5 e^{-x}-x}}{x^2} \, dx-10 \int \frac {e^{10 e^{-x}-x}}{x^2} \, dx-\int \frac {6+x^3}{x^3} \, dx\\ &=-e^{e^x}-2 \int \frac {e^{10 e^{-x}}}{x^3} \, dx+4 \int \frac {e^{5 e^{-x}}}{x^3} \, dx+10 \int \frac {e^{5 e^{-x}-x}}{x^2} \, dx-10 \int \frac {e^{10 e^{-x}-x}}{x^2} \, dx-\int \left (1+\frac {6}{x^3}\right ) \, dx\\ &=-e^{e^x}+\frac {3}{x^2}-x-2 \int \frac {e^{10 e^{-x}}}{x^3} \, dx+4 \int \frac {e^{5 e^{-x}}}{x^3} \, dx+10 \int \frac {e^{5 e^{-x}-x}}{x^2} \, dx-10 \int \frac {e^{10 e^{-x}-x}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.30, size = 43, normalized size = 1.30 \begin {gather*} -e^{e^x}+\frac {3}{x^2}-\frac {2 e^{5 e^{-x}}}{x^2}+\frac {e^{10 e^{-x}}}{x^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(10/E^x)*(-2*E^x - 10*x) - E^(E^x + 2*x)*x^3 + E^(5/E^x)*(4*E^x + 10*x) + E^x*(-6 - x^3))/(E^x*x^
3),x]

[Out]

-E^E^x + 3/x^2 - (2*E^(5/E^x))/x^2 + E^(10/E^x)/x^2 - x

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fricas [B]  time = 0.71, size = 57, normalized size = 1.73 \begin {gather*} -\frac {{\left (x^{2} e^{\left (2 \, x + e^{x}\right )} + {\left (x^{3} - 3\right )} e^{\left (2 \, x\right )} - e^{\left (2 \, x + 10 \, e^{\left (-x\right )}\right )} + 2 \, e^{\left (2 \, x + 5 \, e^{\left (-x\right )}\right )}\right )} e^{\left (-2 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="fricas")

[Out]

-(x^2*e^(2*x + e^x) + (x^3 - 3)*e^(2*x) - e^(2*x + 10*e^(-x)) + 2*e^(2*x + 5*e^(-x)))*e^(-2*x)/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{3} e^{\left (2 \, x + e^{x}\right )} + {\left (x^{3} + 6\right )} e^{x} + 2 \, {\left (5 \, x + e^{x}\right )} e^{\left (10 \, e^{\left (-x\right )}\right )} - 2 \, {\left (5 \, x + 2 \, e^{x}\right )} e^{\left (5 \, e^{\left (-x\right )}\right )}\right )} e^{\left (-x\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="giac")

[Out]

integrate(-(x^3*e^(2*x + e^x) + (x^3 + 6)*e^x + 2*(5*x + e^x)*e^(10*e^(-x)) - 2*(5*x + 2*e^x)*e^(5*e^(-x)))*e^
(-x)/x^3, x)

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maple [A]  time = 0.04, size = 38, normalized size = 1.15

method	result	size
risch	\(-x +\frac {3}{x^{2}}-{\mathrm e}^{{\mathrm e}^{x}}+\frac {{\mathrm e}^{10 \,{\mathrm e}^{-x}}}{x^{2}}-\frac {2 \,{\mathrm e}^{5 \,{\mathrm e}^{-x}}}{x^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*exp(x))
/exp(x)/x^3,x,method=_RETURNVERBOSE)

[Out]

-x+3/x^2-exp(exp(x))+1/x^2*exp(10*exp(-x))-2/x^2*exp(5*exp(-x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x + \frac {3}{x^{2}} - e^{\left (e^{x}\right )} - \int \frac {2 \, {\left (5 \, x + e^{x}\right )} e^{\left (-x + 10 \, e^{\left (-x\right )}\right )}}{x^{3}}\,{d x} + \int \frac {2 \, {\left (5 \, x + 2 \, e^{x}\right )} e^{\left (-x + 5 \, e^{\left (-x\right )}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^3*exp(x)^2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))^2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x^3-6)*e
xp(x))/exp(x)/x^3,x, algorithm="maxima")

[Out]

-x + 3/x^2 - e^(e^x) - integrate(2*(5*x + e^x)*e^(-x + 10*e^(-x))/x^3, x) + integrate(2*(5*x + 2*e^x)*e^(-x +
5*e^(-x))/x^3, x)

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mupad [B]  time = 5.16, size = 37, normalized size = 1.12 \begin {gather*} \frac {{\mathrm {e}}^{10\,{\mathrm {e}}^{-x}}}{x^2}-{\mathrm {e}}^{{\mathrm {e}}^x}-\frac {2\,{\mathrm {e}}^{5\,{\mathrm {e}}^{-x}}}{x^2}-x+\frac {3}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x)*(exp(x)*(x^3 + 6) - exp(5*exp(-x))*(10*x + 4*exp(x)) + exp(10*exp(-x))*(10*x + 2*exp(x)) + x^3*e
xp(2*x)*exp(exp(x))))/x^3,x)

[Out]

exp(10*exp(-x))/x^2 - exp(exp(x)) - (2*exp(5*exp(-x)))/x^2 - x + 3/x^2

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sympy [A]  time = 0.35, size = 36, normalized size = 1.09 \begin {gather*} - x - e^{e^{x}} + \frac {3}{x^{2}} + \frac {x^{2} e^{10 e^{- x}} - 2 x^{2} e^{5 e^{- x}}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**3*exp(x)**2*exp(exp(x))+(-2*exp(x)-10*x)*exp(5/exp(x))**2+(4*exp(x)+10*x)*exp(5/exp(x))+(-x**3-
6)*exp(x))/exp(x)/x**3,x)

[Out]

-x - exp(exp(x)) + 3/x**2 + (x**2*exp(10*exp(-x)) - 2*x**2*exp(5*exp(-x)))/x**4

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