3.40.40 \(\int \frac {e^{-\frac {24}{x^2}} (96+2 x^2)}{x^2} \, dx\)

Optimal. Leaf size=21 \[ -e^5+3 e^9+2 e^{-\frac {24}{x^2}} x \]

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Rubi [A]  time = 0.04, antiderivative size = 10, normalized size of antiderivative = 0.48, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2288} \begin {gather*} 2 e^{-\frac {24}{x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(96 + 2*x^2)/(E^(24/x^2)*x^2),x]

[Out]

(2*x)/E^(24/x^2)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 e^{-\frac {24}{x^2}} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 0.48 \begin {gather*} 2 e^{-\frac {24}{x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96 + 2*x^2)/(E^(24/x^2)*x^2),x]

[Out]

(2*x)/E^(24/x^2)

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fricas [A]  time = 0.63, size = 9, normalized size = 0.43 \begin {gather*} 2 \, x e^{\left (-\frac {24}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+96)/x^2/exp(24/x^2),x, algorithm="fricas")

[Out]

2*x*e^(-24/x^2)

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giac [A]  time = 0.12, size = 9, normalized size = 0.43 \begin {gather*} 2 \, x e^{\left (-\frac {24}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+96)/x^2/exp(24/x^2),x, algorithm="giac")

[Out]

2*x*e^(-24/x^2)

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maple [A]  time = 0.03, size = 10, normalized size = 0.48

method	result	size
derivativedivides	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(10\)
default	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(10\)
risch	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(10\)
gosper	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(12\)
norman	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(12\)
meijerg	\(-2 \sqrt {6}\, \left (-\frac {x \sqrt {6}\, {\mathrm e}^{-\frac {24}{x^{2}}}}{6}-2 \sqrt {\pi }\, \erf \left (\frac {2 \sqrt {6}}{x}\right )\right )-4 \sqrt {6}\, \sqrt {\pi }\, \erf \left (\frac {2 \sqrt {6}}{x}\right )\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+96)/x^2/exp(24/x^2),x,method=_RETURNVERBOSE)

[Out]

2/exp(1/x^2)^24*x

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maxima [C]  time = 0.37, size = 48, normalized size = 2.29 \begin {gather*} 2 \, \sqrt {6} x \sqrt {\frac {1}{x^{2}}} \Gamma \left (-\frac {1}{2}, \frac {24}{x^{2}}\right ) - \frac {4 \, \sqrt {6} \sqrt {\pi } \sqrt {x^{2}} {\left (\operatorname {erf}\left (2 \, \sqrt {6} \sqrt {\frac {1}{x^{2}}}\right ) - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+96)/x^2/exp(24/x^2),x, algorithm="maxima")

[Out]

2*sqrt(6)*x*sqrt(x^(-2))*gamma(-1/2, 24/x^2) - 4*sqrt(6)*sqrt(pi)*sqrt(x^2)*(erf(2*sqrt(6)*sqrt(x^(-2))) - 1)/
x

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mupad [B]  time = 2.21, size = 9, normalized size = 0.43 \begin {gather*} 2\,x\,{\mathrm {e}}^{-\frac {24}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-24/x^2)*(2*x^2 + 96))/x^2,x)

[Out]

2*x*exp(-24/x^2)

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sympy [A]  time = 0.10, size = 8, normalized size = 0.38 \begin {gather*} 2 x e^{- \frac {24}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+96)/x**2/exp(24/x**2),x)

[Out]

2*x*exp(-24/x**2)

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