3.1393 \(\int \frac{1}{x^4 \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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Rubi [A]  time = 0.0030968, antiderivative size = 16, 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 = {264} \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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{1}{x^4 \sqrt{2+x^6}} \, dx &=-\frac{\sqrt{2+x^6}}{6 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0022609, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^6+2}}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(6*x^3)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{3}}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(x^6+2)^(1/2),x)

[Out]

-1/6*(x^6+2)^(1/2)/x^3

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Maxima [A]  time = 1.01833, size = 16, normalized size = 1. \begin{align*} -\frac{\sqrt{x^{6} + 2}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(x^6 + 2)/x^3

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Fricas [A]  time = 1.57111, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{3} + \sqrt{x^{6} + 2}}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(x^3 + sqrt(x^6 + 2))/x^3

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Sympy [A]  time = 0.588745, size = 12, normalized size = 0.75 \begin{align*} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(x**6+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**6)/6

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Giac [A]  time = 1.19195, size = 27, normalized size = 1.69 \begin{align*} -\frac{\sqrt{\frac{2}{x^{6}} + 1}}{6 \, \mathrm{sgn}\left (x\right )} + \frac{1}{6} \, \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

-1/6*sqrt(2/x^6 + 1)/sgn(x) + 1/6*sgn(x)