3.939 \(\int \frac{x^3}{(1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{2 \sqrt{x^4+1}} \]

[Out]

-1/(2*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0026001, 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} \[ -\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + x^4)^(3/2),x]

[Out]

-1/(2*Sqrt[1 + x^4])

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 )^{3/2}} \, dx &=-\frac{1}{2 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0020503, size = 13, normalized size = 1. \[ -\frac{1}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + x^4)^(3/2),x]

[Out]

-1/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.01, size = 10, normalized size = 0.8 \begin{align*} -{\frac{1}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.997663, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

-1/2/sqrt(x^4 + 1)

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Fricas [A]  time = 1.42802, size = 27, normalized size = 2.08 \begin{align*} -\frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/2/sqrt(x^4 + 1)

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Sympy [A]  time = 0.46822, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+1)**(3/2),x)

[Out]

-1/(2*sqrt(x**4 + 1))

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Giac [A]  time = 1.17582, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

-1/2/sqrt(x^4 + 1)