\(\int \frac {x^3}{(a^4+x^4)^3} \, dx\) [172]

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]

[In]

Int[x^3/(a^4 + x^4)^3,x]

[Out]

-1/8*1/(a^4 + x^4)^2

Rule 267

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*} \text {integral}& = -\frac {1}{8 \left (a^4+x^4\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \left (a^4+x^4\right )^2} \]

[In]

Integrate[x^3/(a^4 + x^4)^3,x]

[Out]

-1/8*1/(a^4 + x^4)^2

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 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	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)
parallelrisch	\(-\frac {1}{8 \left (a^{4}+x^{4}\right )^{2}}\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )}} \]

[In]

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

[Out]

-1/8/(a^8 + 2*a^4*x^4 + x^8)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=- \frac {1}{8 a^{8} + 16 a^{4} x^{4} + 8 x^{8}} \]

[In]

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

[Out]

-1/(8*a**8 + 16*a**4*x**4 + 8*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]

[In]

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

[Out]

-1/8/(a^4 + x^4)^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8 \, {\left (a^{4} + x^{4}\right )}^{2}} \]

[In]

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

[Out]

-1/8/(a^4 + x^4)^2

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{\left (a^4+x^4\right )^3} \, dx=-\frac {1}{8\,{\left (a^4+x^4\right )}^2} \]

[In]

int(x^3/(a^4 + x^4)^3,x)

[Out]

-1/(8*(a^4 + x^4)^2)