3.172 \(\int \frac{x^3}{(a^4+x^4)^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{8 \left (a^4+x^4\right )^2} \]

[Out]

-1/(8*(a^4 + x^4)^2)

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Rubi [A]  time = 0.0023815, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{8 \left (a^4+x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a^4 + x^4)^3,x]

[Out]

-1/(8*(a^4 + x^4)^2)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x^3}{\left (a^4+x^4\right )^3} \, dx &=-\frac{1}{8 \left (a^4+x^4\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0034365, size = 13, normalized size = 1. \[ -\frac{1}{8 \left (a^4+x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a^4 + x^4)^3,x]

[Out]

-1/(8*(a^4 + x^4)^2)

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Maple [A]  time = 0., size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{8\, \left ({a}^{4}+{x}^{4} \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a^4+x^4)^3,x)

[Out]

-1/8/(a^4+x^4)^2

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Maxima [A]  time = 0.921507, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{8 \,{\left (a^{4} + x^{4}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a^4+x^4)^3,x, algorithm="maxima")

[Out]

-1/8/(a^4 + x^4)^2

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Fricas [A]  time = 1.87433, size = 41, normalized size = 3.15 \begin{align*} -\frac{1}{8 \,{\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a^4+x^4)^3,x, algorithm="fricas")

[Out]

-1/8/(a^8 + 2*a^4*x^4 + x^8)

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Sympy [A]  time = 0.795548, size = 20, normalized size = 1.54 \begin{align*} - \frac{1}{8 a^{8} + 16 a^{4} x^{4} + 8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a**4+x**4)**3,x)

[Out]

-1/(8*a**8 + 16*a**4*x**4 + 8*x**8)

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Giac [A]  time = 1.0583, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{8 \,{\left (a^{4} + x^{4}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a^4+x^4)^3,x, algorithm="giac")

[Out]

-1/8/(a^4 + x^4)^2