\(\int \frac {e^{-3+x^4} (-1+4 x^4)}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx\) [9206]

Optimal result

Integrand size = 40, antiderivative size = 19 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=\frac {x}{x+e^{3-x^4} x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6820, 6843, 32} \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {e^3}{\frac {e^{x^4}}{x}+e^3} \]

[In]

Int[(E^(-3 + x^4)*(-1 + 4*x^4))/(E^(-6 + 2*x^4) + 2*E^(-3 + x^4)*x + x^2),x]

[Out]

-(E^3/(E^3 + E^x^4/x))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6843

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{3+x^4} \left (-1+4 x^4\right )}{\left (e^{x^4}+e^3 x\right )^2} \, dx \\ & = e^3 \text {Subst}\left (\int \frac {1}{\left (e^3+x\right )^2} \, dx,x,\frac {e^{x^4}}{x}\right ) \\ & = -\frac {e^3}{e^3+\frac {e^{x^4}}{x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {e^3 x}{e^{x^4}+e^3 x} \]

[In]

Integrate[(E^(-3 + x^4)*(-1 + 4*x^4))/(E^(-6 + 2*x^4) + 2*E^(-3 + x^4)*x + x^2),x]

[Out]

-((E^3*x)/(E^x^4 + E^3*x))

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {x}{{\mathrm e}^{x^{4}-3}+x}\)	\(14\)
parallelrisch	\(-\frac {x}{{\mathrm e}^{x^{4}-3}+x}\)	\(14\)
norman	\(\frac {{\mathrm e}^{x^{4}-3}}{{\mathrm e}^{x^{4}-3}+x}\)	\(18\)

[In]

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

[Out]

-x/(exp(x^4-3)+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {x}{x + e^{\left (x^{4} - 3\right )}} \]

[In]

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

[Out]

-x/(x + e^(x^4 - 3))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.53 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=- \frac {x}{x + e^{x^{4} - 3}} \]

[In]

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

[Out]

-x/(x + exp(x**4 - 3))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {x e^{3}}{x e^{3} + e^{\left (x^{4}\right )}} \]

[In]

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

[Out]

-x*e^3/(x*e^3 + e^(x^4))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {x e^{3}}{x e^{3} + e^{\left (x^{4}\right )}} \]

[In]

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

[Out]

-x*e^3/(x*e^3 + e^(x^4))

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3+x^4} \left (-1+4 x^4\right )}{e^{-6+2 x^4}+2 e^{-3+x^4} x+x^2} \, dx=-\frac {x}{x+{\mathrm {e}}^{x^4-3}} \]

[In]

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

[Out]

-x/(x + exp(x^4 - 3))