3.52.14 \(\int \frac {10 e^{e^{5/x}+\frac {5}{x}} x-x^3+e^{-2+e^{5/x}+e^x+2 x} (-5 e^{5/x}-x+2 x^2+e^x x^2)}{x^3} \, dx\) [5114]

Optimal. Leaf size=33 \[ 7-e^{e^{5/x}} \left (2-\frac {e^{-2+e^x+2 x}}{x}\right )-x \]

[Out]

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

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Rubi [F]
time = 1.25, 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 {10 e^{e^{5/x}+\frac {5}{x}} x-x^3+e^{-2+e^{5/x}+e^x+2 x} \left (-5 e^{5/x}-x+2 x^2+e^x x^2\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 36, normalized size = 1.09 \begin {gather*} -2 e^{e^{5/x}}+\frac {e^{-2+e^{5/x}+e^x+2 x}}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.27, size = 32, normalized size = 0.97

method	result	size
risch	\(-x -2 \,{\mathrm e}^{{\mathrm e}^{\frac {5}{x}}}+\frac {{\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+{\mathrm e}^{x}+2 x -2}}{x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*x^2-5*exp(5/x)+2*x^2-x)*exp(exp(5/x))*exp(exp(x)+2*x-2)+10*x*exp(5/x)*exp(exp(5/x))-x^3)/x^3,x,me
thod=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 31, normalized size = 0.94 \begin {gather*} -x + \frac {e^{\left (2 \, x + e^{x} + e^{\frac {5}{x}} - 2\right )}}{x} - 2 \, e^{\left (e^{\frac {5}{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).
time = 0.34, size = 61, normalized size = 1.85 \begin {gather*} -\frac {{\left (x^{2} e^{\frac {5}{x}} + 2 \, x e^{\left (\frac {x e^{\frac {5}{x}} + 5}{x}\right )} - e^{\left (2 \, x + \frac {5}{x} + e^{x} + e^{\frac {5}{x}} - 2\right )}\right )} e^{\left (-\frac {5}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.26, size = 27, normalized size = 0.82 \begin {gather*} - x - 2 e^{e^{\frac {5}{x}}} + \frac {e^{2 x + e^{x} - 2} e^{e^{\frac {5}{x}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (27) = 54\).
time = 0.43, size = 69, normalized size = 2.09 \begin {gather*} -\frac {{\left (x^{2} e^{\frac {5}{x}} + 2 \, x e^{\left (\frac {x e^{\frac {5}{x}} + 5}{x}\right )} - e^{\left (\frac {2 \, x^{2} + x e^{x} + x e^{\frac {5}{x}} - 2 \, x + 5}{x}\right )}\right )} e^{\left (-\frac {5}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.37, size = 30, normalized size = 0.91 \begin {gather*} -x-\frac {{\mathrm {e}}^{{\mathrm {e}}^{5/x}}\,\left (2\,x-{\mathrm {e}}^{2\,x+{\mathrm {e}}^x-2}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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