3.20.28 \(\int \frac {e^{\frac {e+4 x^2}{x^2}} (-6 e+3 x^2)}{x^2} \, dx\)

Optimal. Leaf size=12 \[ 3 e^{4+\frac {e}{x^2}} x \]

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Rubi [B]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 3.33, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2288} \begin {gather*} -\frac {3 e^{\frac {4 x^2+e}{x^2}+1}}{x^2 \left (\frac {4}{x}-\frac {4 x^2+e}{x^3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((E + 4*x^2)/x^2)*(-6*E + 3*x^2))/x^2,x]

[Out]

(-3*E^(1 + (E + 4*x^2)/x^2))/(x^2*(4/x - (E + 4*x^2)/x^3))

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} &=-\frac {3 e^{1+\frac {e+4 x^2}{x^2}}}{x^2 \left (\frac {4}{x}-\frac {e+4 x^2}{x^3}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} 3 e^{4+\frac {e}{x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E + 4*x^2)/x^2)*(-6*E + 3*x^2))/x^2,x]

[Out]

3*E^(4 + E/x^2)*x

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fricas [A]  time = 0.94, size = 16, normalized size = 1.33 \begin {gather*} 3 \, x e^{\left (\frac {4 \, x^{2} + e}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*exp(1)+3*x^2)*exp((exp(1)+4*x^2)/x^2)/x^2,x, algorithm="fricas")

[Out]

3*x*e^((4*x^2 + e)/x^2)

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giac [A]  time = 0.36, size = 16, normalized size = 1.33 \begin {gather*} 3 \, x e^{\left (\frac {4 \, x^{2} + e}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*exp(1)+3*x^2)*exp((exp(1)+4*x^2)/x^2)/x^2,x, algorithm="giac")

[Out]

3*x*e^((4*x^2 + e)/x^2)

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maple [A]  time = 0.08, size = 17, normalized size = 1.42

method	result	size
gosper	\(3 \,{\mathrm e}^{\frac {{\mathrm e}+4 x^{2}}{x^{2}}} x\)	\(17\)
norman	\(3 \,{\mathrm e}^{\frac {{\mathrm e}+4 x^{2}}{x^{2}}} x\)	\(17\)
risch	\(3 \,{\mathrm e}^{\frac {{\mathrm e}+4 x^{2}}{x^{2}}} x\)	\(17\)
meijerg	\(3 \,{\mathrm e}^{\frac {9}{2}} \sqrt {\pi }\, \erfi \left (\frac {{\mathrm e}^{\frac {1}{2}}}{x}\right )-\frac {3 i {\mathrm e}^{\frac {9}{2}} \left (2 i x \,{\mathrm e}^{-\frac {1}{2}+\frac {{\mathrm e}}{x^{2}}}-2 i \sqrt {\pi }\, \erfi \left (\frac {{\mathrm e}^{\frac {1}{2}}}{x}\right )\right )}{2}\)	\(48\)
derivativedivides	\(-3 \,{\mathrm e}^{4} \left (-x \,{\mathrm e}^{\frac {{\mathrm e}}{x^{2}}}-i {\mathrm e}^{\frac {1}{2}} \sqrt {\pi }\, \erf \left (\frac {i {\mathrm e}^{\frac {1}{2}}}{x}\right )\right )-3 i {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{4} \sqrt {\pi }\, \erf \left (\frac {i {\mathrm e}^{\frac {1}{2}}}{x}\right )\)	\(61\)
default	\(-3 \,{\mathrm e}^{4} \left (-x \,{\mathrm e}^{\frac {{\mathrm e}}{x^{2}}}-i {\mathrm e}^{\frac {1}{2}} \sqrt {\pi }\, \erf \left (\frac {i {\mathrm e}^{\frac {1}{2}}}{x}\right )\right )-3 i {\mathrm e}^{\frac {1}{2}} {\mathrm e}^{4} \sqrt {\pi }\, \erf \left (\frac {i {\mathrm e}^{\frac {1}{2}}}{x}\right )\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*exp(1)+3*x^2)*exp((exp(1)+4*x^2)/x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

3*exp((exp(1)+4*x^2)/x^2)*x

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maxima [C]  time = 0.86, size = 55, normalized size = 4.58 \begin {gather*} \frac {3}{2} \, x \sqrt {-\frac {e}{x^{2}}} e^{4} \Gamma \left (-\frac {1}{2}, -\frac {e}{x^{2}}\right ) + \frac {3 \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-\frac {e}{x^{2}}}\right ) - 1\right )} e^{5}}{x \sqrt {-\frac {e}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*exp(1)+3*x^2)*exp((exp(1)+4*x^2)/x^2)/x^2,x, algorithm="maxima")

[Out]

3/2*x*sqrt(-e/x^2)*e^4*gamma(-1/2, -e/x^2) + 3*sqrt(pi)*(erf(sqrt(-e/x^2)) - 1)*e^5/(x*sqrt(-e/x^2))

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mupad [B]  time = 1.13, size = 12, normalized size = 1.00 \begin {gather*} 3\,x\,{\mathrm {e}}^{\frac {\mathrm {e}}{x^2}}\,{\mathrm {e}}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(1) + 4*x^2)/x^2)*(6*exp(1) - 3*x^2))/x^2,x)

[Out]

3*x*exp(exp(1)/x^2)*exp(4)

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sympy [A]  time = 0.13, size = 15, normalized size = 1.25 \begin {gather*} 3 x e^{\frac {4 x^{2} + e}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*exp(1)+3*x**2)*exp((exp(1)+4*x**2)/x**2)/x**2,x)

[Out]

3*x*exp((4*x**2 + E)/x**2)

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