\(\int \frac {e^{-\frac {24}{x^2}} (96+2 x^2)}{x^2} \, dx\) [3940]

Optimal result

Integrand size = 18, antiderivative size = 21 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=-e^5+3 e^9+2 e^{-\frac {24}{x^2}} x \]

[Out]

3*exp(9)+2*x/exp(24/x^2)-exp(5)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2326} \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 e^{-\frac {24}{x^2}} x \]

[In]

Int[(96 + 2*x^2)/(E^(24/x^2)*x^2),x]

[Out]

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

Rule 2326

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 e^{-\frac {24}{x^2}} x \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 10, normalized size of antiderivative = 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	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(12\)
parallelrisch	\(2 x \,{\mathrm e}^{-\frac {24}{x^{2}}}\)	\(12\)
meijerg	\(-2 \sqrt {6}\, \left (-\frac {x \sqrt {6}\, {\mathrm e}^{-\frac {24}{x^{2}}}}{6}-2 \sqrt {\pi }\, \operatorname {erf}\left (\frac {2 \sqrt {6}}{x}\right )\right )-4 \sqrt {6}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {2 \sqrt {6}}{x}\right )\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 \, x e^{\left (-\frac {24}{x^{2}}\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.38 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 x e^{- \frac {24}{x^{2}}} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 \, \sqrt {6} x \sqrt {\frac {1}{x^{2}}} \Gamma \left (-\frac {1}{2}, \frac {24}{x^{2}}\right ) - \frac {4 \, \sqrt {6} \sqrt {\pi } \sqrt {x^{2}} {\left (\operatorname {erf}\left (2 \, \sqrt {6} \sqrt {\frac {1}{x^{2}}}\right ) - 1\right )}}{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2 \, x e^{\left (-\frac {24}{x^{2}}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.71 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\frac {24}{x^2}} \left (96+2 x^2\right )}{x^2} \, dx=2\,x\,{\mathrm {e}}^{-\frac {24}{x^2}} \]

[In]

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

[Out]

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