3.89.58 \(\int \frac {e^{\frac {e^{\frac {2}{x^2}} x+x^2+\log (\frac {5}{3})}{x}} (-4 e^{\frac {2}{x^2}}+x^3-x \log (\frac {5}{3}))}{x^3} \, dx\)

Optimal. Leaf size=19 \[ e^{e^{\frac {2}{x^2}}+x+\frac {\log \left (\frac {5}{3}\right )}{x}} \]

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Rubi [A]  time = 0.57, antiderivative size = 27, normalized size of antiderivative = 1.42, number of steps used = 1, number of rules used = 1, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6706} \begin {gather*} \left (\frac {5}{3}\right )^{\frac {1}{x}} e^{\frac {x^2+e^{\frac {2}{x^2}} x}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (\frac {5}{3}\right )^{\frac {1}{x}} e^{\frac {e^{\frac {2}{x^2}} x+x^2}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 1.00 \begin {gather*} \left (\frac {5}{3}\right )^{\frac {1}{x}} e^{e^{\frac {2}{x^2}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5/3)^x^(-1)*E^(E^(2/x^2) + x)

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fricas [A]  time = 0.51, size = 21, normalized size = 1.11 \begin {gather*} e^{\left (\frac {x^{2} + x e^{\left (\frac {2}{x^{2}}\right )} - \log \left (\frac {3}{5}\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2/x^2)+x*log(3/5)+x^3)*exp((x*exp(2/x^2)-log(3/5)+x^2)/x)/x^3,x, algorithm="fricas")

[Out]

e^((x^2 + x*e^(2/x^2) - log(3/5))/x)

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giac [A]  time = 0.15, size = 16, normalized size = 0.84 \begin {gather*} e^{\left (x - \frac {\log \left (\frac {3}{5}\right )}{x} + e^{\left (\frac {2}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2/x^2)+x*log(3/5)+x^3)*exp((x*exp(2/x^2)-log(3/5)+x^2)/x)/x^3,x, algorithm="giac")

[Out]

e^(x - log(3/5)/x + e^(2/x^2))

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maple [A]  time = 0.09, size = 21, normalized size = 1.11

method	result	size
risch	\(5^{\frac {1}{x}} \left (\frac {1}{3}\right )^{\frac {1}{x}} {\mathrm e}^{{\mathrm e}^{\frac {2}{x^{2}}}+x}\)	\(21\)
norman	\({\mathrm e}^{\frac {x \,{\mathrm e}^{\frac {2}{x^{2}}}-\ln \left (\frac {3}{5}\right )+x^{2}}{x}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 22, normalized size = 1.16 \begin {gather*} e^{\left (x + \frac {\log \relax (5)}{x} - \frac {\log \relax (3)}{x} + e^{\left (\frac {2}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(2/x^2)+x*log(3/5)+x^3)*exp((x*exp(2/x^2)-log(3/5)+x^2)/x)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.25, size = 15, normalized size = 0.79 \begin {gather*} {\left (\frac {5}{3}\right )}^{1/x}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {2}{x^2}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x*exp(2/x^2) - log(3/5) + x^2)/x)*(x*log(3/5) - 4*exp(2/x^2) + x^3))/x^3,x)

[Out]

(5/3)^(1/x)*exp(exp(2/x^2))*exp(x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: IndexError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

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