3.90.38 \(\int \frac {e^{e^x+x} x^3+e^{e^{\frac {9}{x^2}+\frac {5}{x}}+e^{\frac {9}{x^2}} x} (e^{\frac {9}{x^2}+\frac {5}{x}} (-18-5 x)+e^{\frac {9}{x^2}} (-18 x+x^3))}{x^3} \, dx\)

Optimal. Leaf size=25 \[ e^{e^x}+e^{e^{\frac {9}{x^2}} \left (e^{5/x}+x\right )} \]

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Rubi [F]  time = 2.53, 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^{e^x+x} x^3+e^{e^{\frac {9}{x^2}+\frac {5}{x}}+e^{\frac {9}{x^2}} x} \left (e^{\frac {9}{x^2}+\frac {5}{x}} (-18-5 x)+e^{\frac {9}{x^2}} \left (-18 x+x^3\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^x + x)*x^3 + E^(E^(9/x^2 + 5/x) + E^(9/x^2)*x)*(E^(9/x^2 + 5/x)*(-18 - 5*x) + E^(9/x^2)*(-18*x + x^3
)))/x^3,x]

[Out]

E^E^x + Defer[Int][E^(E^(9/x^2 + 5/x) + 9/x^2 + E^(9/x^2)*x), x] - 18*Defer[Int][E^(E^(9/x^2 + 5/x) + 9/x^2 +
5/x + E^(9/x^2)*x)/x^3, x] - 18*Defer[Int][E^(E^(9/x^2 + 5/x) + 9/x^2 + E^(9/x^2)*x)/x^2, x] - 5*Defer[Int][E^
(E^(9/x^2 + 5/x) + 9/x^2 + 5/x + E^(9/x^2)*x)/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{e^x+x}+\frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x} \left (-18 e^{5/x}-18 x-5 e^{5/x} x+x^3\right )}{x^3}\right ) \, dx\\ &=\int e^{e^x+x} \, dx+\int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x} \left (-18 e^{5/x}-18 x-5 e^{5/x} x+x^3\right )}{x^3} \, dx\\ &=\int \left (-\frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x} (18+5 x)}{x^3}+\frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x} \left (-18+x^2\right )}{x^2}\right ) \, dx+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}-\int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x} (18+5 x)}{x^3} \, dx+\int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x} \left (-18+x^2\right )}{x^2} \, dx\\ &=e^{e^x}+\int \left (e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x}-\frac {18 e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x}}{x^2}\right ) \, dx-\int \left (\frac {18 e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x}}{x^3}+\frac {5 e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x}}{x^2}\right ) \, dx\\ &=e^{e^x}-5 \int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x}}{x^2} \, dx-18 \int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+\frac {5}{x}+e^{\frac {9}{x^2}} x}}{x^3} \, dx-18 \int \frac {e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x}}{x^2} \, dx+\int e^{e^{\frac {9}{x^2}+\frac {5}{x}}+\frac {9}{x^2}+e^{\frac {9}{x^2}} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 31, normalized size = 1.24 \begin {gather*} e^{e^x}+e^{e^{\frac {9}{x^2}+\frac {5}{x}}+e^{\frac {9}{x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^x + x)*x^3 + E^(E^(9/x^2 + 5/x) + E^(9/x^2)*x)*(E^(9/x^2 + 5/x)*(-18 - 5*x) + E^(9/x^2)*(-18*x
+ x^3)))/x^3,x]

[Out]

E^E^x + E^(E^(9/x^2 + 5/x) + E^(9/x^2)*x)

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fricas [A]  time = 0.52, size = 32, normalized size = 1.28 \begin {gather*} {\left (e^{\left (x e^{\left (\frac {9}{x^{2}}\right )} + x + e^{\left (\frac {5 \, x + 9}{x^{2}}\right )}\right )} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3*exp(x)*exp(exp(x))+((-5*x-18)*exp(9/x^2)*exp(5/x)+(x^3-18*x)*exp(9/x^2))*exp(exp(9/x^2)*exp(5/x
)+x*exp(9/x^2)))/x^3,x, algorithm="fricas")

[Out]

(e^(x*e^(9/x^2) + x + e^((5*x + 9)/x^2)) + e^(x + e^x))*e^(-x)

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giac [B]  time = 0.20, size = 55, normalized size = 2.20 \begin {gather*} {\left (e^{\left (x + \frac {x^{3} e^{\left (\frac {9}{x^{2}}\right )} + x^{2} e^{\left (\frac {5 \, x + 9}{x^{2}}\right )} + 9}{x^{2}}\right )} + e^{\left (x + \frac {9}{x^{2}} + e^{x}\right )}\right )} e^{\left (-x - \frac {9}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3*exp(x)*exp(exp(x))+((-5*x-18)*exp(9/x^2)*exp(5/x)+(x^3-18*x)*exp(9/x^2))*exp(exp(9/x^2)*exp(5/x
)+x*exp(9/x^2)))/x^3,x, algorithm="giac")

[Out]

(e^(x + (x^3*e^(9/x^2) + x^2*e^((5*x + 9)/x^2) + 9)/x^2) + e^(x + 9/x^2 + e^x))*e^(-x - 9/x^2)

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maple [A]  time = 0.15, size = 25, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{\frac {5 x +9}{x^{2}}}+x \,{\mathrm e}^{\frac {9}{x^{2}}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*exp(x)*exp(exp(x))+((-5*x-18)*exp(9/x^2)*exp(5/x)+(x^3-18*x)*exp(9/x^2))*exp(exp(9/x^2)*exp(5/x)+x*ex
p(9/x^2)))/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(exp(x))+exp(exp((5*x+9)/x^2)+x*exp(9/x^2))

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maxima [A]  time = 0.47, size = 26, normalized size = 1.04 \begin {gather*} e^{\left (x e^{\left (\frac {9}{x^{2}}\right )} + e^{\left (\frac {5}{x} + \frac {9}{x^{2}}\right )}\right )} + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3*exp(x)*exp(exp(x))+((-5*x-18)*exp(9/x^2)*exp(5/x)+(x^3-18*x)*exp(9/x^2))*exp(exp(9/x^2)*exp(5/x
)+x*exp(9/x^2)))/x^3,x, algorithm="maxima")

[Out]

e^(x*e^(9/x^2) + e^(5/x + 9/x^2)) + e^(e^x)

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mupad [B]  time = 7.88, size = 28, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}+{\mathrm {e}}^{x\,{\mathrm {e}}^{\frac {9}{x^2}}}\,{\mathrm {e}}^{{\mathrm {e}}^{5/x}\,{\mathrm {e}}^{\frac {9}{x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*exp(9/x^2) + exp(5/x)*exp(9/x^2))*(exp(9/x^2)*(18*x - x^3) + exp(5/x)*exp(9/x^2)*(5*x + 18)) - x^3
*exp(exp(x))*exp(x))/x^3,x)

[Out]

exp(exp(x)) + exp(x*exp(9/x^2))*exp(exp(5/x)*exp(9/x^2))

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sympy [A]  time = 3.43, size = 26, normalized size = 1.04 \begin {gather*} e^{x e^{\frac {9}{x^{2}}} + e^{\frac {9}{x^{2}}} e^{\frac {5}{x}}} + e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3*exp(x)*exp(exp(x))+((-5*x-18)*exp(9/x**2)*exp(5/x)+(x**3-18*x)*exp(9/x**2))*exp(exp(9/x**2)*ex
p(5/x)+x*exp(9/x**2)))/x**3,x)

[Out]

exp(x*exp(9/x**2) + exp(9/x**2)*exp(5/x)) + exp(exp(x))

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