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

Optimal. Leaf size=16 \[ \frac{x^2}{2 \sqrt{x^4+1}} \]

[Out]

x^2/(2*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0022335, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {264} \[ \frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0021165, size = 16, normalized size = 1. \[ \frac{x^2}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 + x^4])

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Maple [A]  time = 0.012, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964204, size = 16, normalized size = 1. \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)

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Fricas [B]  time = 1.44875, size = 61, normalized size = 3.81 \begin{align*} \frac{x^{4} + \sqrt{x^{4} + 1} x^{2} + 1}{2 \,{\left (x^{4} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^4 + sqrt(x^4 + 1)*x^2 + 1)/(x^4 + 1)

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Sympy [A]  time = 0.648047, size = 12, normalized size = 0.75 \begin{align*} \frac{x^{2}}{2 \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(2*sqrt(x**4 + 1))

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Giac [A]  time = 1.21079, size = 16, normalized size = 1. \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/sqrt(x^4 + 1)