3.21.93 \(\int \frac {-4 x^2+e^{2 x^2} (6-8 x^2)}{x^4} \, dx\)

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

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Rubi [A]  time = 0.06, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2288} \begin {gather*} \frac {4}{x}-\frac {2 e^{2 x^2}}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 + E^(2*x^2)*(6 - 8*x^2))/x^4,x]

[Out]

(-2*E^(2*x^2))/x^3 + 4/x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} -\frac {2 e^{2 x^2}}{x^3}+\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + E^(2*x^2)*(6 - 8*x^2))/x^4,x]

[Out]

(-2*E^(2*x^2))/x^3 + 4/x

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fricas [A]  time = 0.89, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - e^{\left (2 \, x^{2}\right )}\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+6)*exp(x^2)^2-4*x^2)/x^4,x, algorithm="fricas")

[Out]

2*(2*x^2 - e^(2*x^2))/x^3

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giac [A]  time = 1.09, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} - e^{\left (2 \, x^{2}\right )}\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+6)*exp(x^2)^2-4*x^2)/x^4,x, algorithm="giac")

[Out]

2*(2*x^2 - e^(2*x^2))/x^3

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maple [A]  time = 0.03, size = 18, normalized size = 0.95

method	result	size
default	\(\frac {4}{x}-\frac {2 \,{\mathrm e}^{2 x^{2}}}{x^{3}}\)	\(18\)
risch	\(\frac {4}{x}-\frac {2 \,{\mathrm e}^{2 x^{2}}}{x^{3}}\)	\(18\)
norman	\(\frac {4 x^{2}-2 \,{\mathrm e}^{2 x^{2}}}{x^{3}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^2+6)*exp(x^2)^2-4*x^2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.53, size = 50, normalized size = 2.63 \begin {gather*} \frac {4 \, \sqrt {2} \sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -2 \, x^{2}\right )}{x} - \frac {6 \, \sqrt {2} \left (-x^{2}\right )^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, -2 \, x^{2}\right )}{x^{3}} + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^2+6)*exp(x^2)^2-4*x^2)/x^4,x, algorithm="maxima")

[Out]

4*sqrt(2)*sqrt(-x^2)*gamma(-1/2, -2*x^2)/x - 6*sqrt(2)*(-x^2)^(3/2)*gamma(-3/2, -2*x^2)/x^3 + 4/x

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mupad [B]  time = 1.18, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2\,\left ({\mathrm {e}}^{2\,x^2}-2\,x^2\right )}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x^2)*(8*x^2 - 6) + 4*x^2)/x^4,x)

[Out]

-(2*(exp(2*x^2) - 2*x^2))/x^3

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sympy [A]  time = 0.09, size = 14, normalized size = 0.74 \begin {gather*} \frac {4}{x} - \frac {2 e^{2 x^{2}}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**2+6)*exp(x**2)**2-4*x**2)/x**4,x)

[Out]

4/x - 2*exp(2*x**2)/x**3

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