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

Optimal. Leaf size=19 \[ \frac {x}{x+e^{3-x^4} x^2} \]

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Rubi [A]  time = 0.35, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6688, 6711, 32} \begin {gather*} -\frac {e^3}{\frac {e^{x^4}}{x}+e^3} \end {gather*}

Antiderivative was successfully verified.

[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 6688

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

Rule 6711

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 {gather*} \begin {aligned} \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 \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 19, normalized size = 1.00 \begin {gather*} -\frac {e^3 x}{e^{x^4}+e^3 x} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.87, size = 13, normalized size = 0.68 \begin {gather*} -\frac {x}{x + e^{\left (x^{4} - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.14, size = 16, normalized size = 0.84 \begin {gather*} -\frac {x e^{3}}{x e^{3} + e^{\left (x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.09, size = 14, normalized size = 0.74

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.45, size = 16, normalized size = 0.84 \begin {gather*} -\frac {x e^{3}}{x e^{3} + e^{\left (x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 7.33, size = 13, normalized size = 0.68 \begin {gather*} -\frac {x}{x+{\mathrm {e}}^{x^4-3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.13, size = 10, normalized size = 0.53 \begin {gather*} - \frac {x}{x + e^{x^{4} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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