\(\int \frac {-4 x^2+e^{2 x^2} (6-8 x^2)}{x^4} \, dx\) [2093]

Optimal result

Integrand size = 25, antiderivative size = 19 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {2 \left (2-\frac {e^{2 x^2}}{x^2}\right )}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2326} \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {4}{x}-\frac {2 e^{2 x^2}}{x^3} \]

[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 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}& = \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{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=-\frac {2 e^{2 x^2}}{x^3}+\frac {4}{x} \]

[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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {2 \, {\left (2 \, x^{2} - e^{\left (2 \, x^{2}\right )}\right )}}{x^{3}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {4}{x} - \frac {2 e^{2 x^{2}}}{x^{3}} \]

[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

Maxima [C] (verification not implemented)

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

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {4 \, \sqrt {2} \sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -2 \, x^{2}\right )}{x} - \frac {6 \, \sqrt {2} \left (-x^{2}\right )^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, -2 \, x^{2}\right )}{x^{3}} + \frac {4}{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=\frac {2 \, {\left (2 \, x^{2} - e^{\left (2 \, x^{2}\right )}\right )}}{x^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 9.57 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-4 x^2+e^{2 x^2} \left (6-8 x^2\right )}{x^4} \, dx=-\frac {2\,\left ({\mathrm {e}}^{2\,x^2}-2\,x^2\right )}{x^3} \]

[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