3.986 \(\int \frac {x^3}{(1+x^4)^{4/3}} \, dx\)

Optimal. Leaf size=13 \[ -\frac {3}{4 \sqrt [3]{x^4+1}} \]

[Out]

-3/4/(x^4+1)^(1/3)

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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} \[ -\frac {3}{4 \sqrt [3]{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + x^4)^(4/3),x]

[Out]

-3/(4*(1 + x^4)^(1/3))

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 (1+x^4\right )^{4/3}} \, dx &=-\frac {3}{4 \sqrt [3]{1+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {3}{4 \sqrt [3]{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + x^4)^(4/3),x]

[Out]

-3/(4*(1 + x^4)^(1/3))

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fricas [A]  time = 1.05, size = 9, normalized size = 0.69 \[ -\frac {3}{4 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(4/3),x, algorithm="fricas")

[Out]

-3/4/(x^4 + 1)^(1/3)

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giac [A]  time = 0.16, size = 9, normalized size = 0.69 \[ -\frac {3}{4 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(4/3),x, algorithm="giac")

[Out]

-3/4/(x^4 + 1)^(1/3)

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maple [A]  time = 0.00, size = 10, normalized size = 0.77 \[ -\frac {3}{4 \left (x^{4}+1\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+1)^(4/3),x)

[Out]

-3/4/(x^4+1)^(1/3)

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maxima [A]  time = 1.35, size = 9, normalized size = 0.69 \[ -\frac {3}{4 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(4/3),x, algorithm="maxima")

[Out]

-3/4/(x^4 + 1)^(1/3)

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mupad [B]  time = 1.10, size = 9, normalized size = 0.69 \[ -\frac {3}{4\,{\left (x^4+1\right )}^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4 + 1)^(4/3),x)

[Out]

-3/(4*(x^4 + 1)^(1/3))

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sympy [A]  time = 0.56, size = 12, normalized size = 0.92 \[ - \frac {3}{4 \sqrt [3]{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+1)**(4/3),x)

[Out]

-3/(4*(x**4 + 1)**(1/3))

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