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

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{8 \left (a^4+x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a^4 + x^4)^3,x]

[Out]

-1/8*1/(a^4 + x^4)^2

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {1}{8 \left (a^4+x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/(a^4 + x^4)^3,x]

[Out]

-1/8*1/(a^4 + x^4)^2

________________________________________________________________________________________

fricas [A]  time = 1.41, size = 19, normalized size = 1.46 \[ -\frac {1}{8 \, {\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.93, size = 11, normalized size = 0.85 \[ -\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.26, size = 12, normalized size = 0.92

method	result	size
gosper	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)
derivativedivides	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)
default	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)
norman	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)
risch	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a^4+x^4)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 11, normalized size = 0.85 \[ -\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 11, normalized size = 0.85 \[ -\frac {1}{8\,{\left (a^4+x^4\right )}^2} \]

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)

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 20, normalized size = 1.54 \[ - \frac {1}{8 a^{8} + 16 a^{4} x^{4} + 8 x^{8}} \]

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)

________________________________________________________________________________________