3.1957 \(\int \frac{\sqrt [3]{1+\frac{1}{x^2}}}{x^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{3}{8} \left (\frac{1}{x^2}+1\right )^{4/3} \]

[Out]

(-3*(1 + x^(-2))^(4/3))/8

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Rubi [A]  time = 0.0031485, 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} \[ -\frac{3}{8} \left (\frac{1}{x^2}+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^(-2))^(1/3)/x^3,x]

[Out]

(-3*(1 + x^(-2))^(4/3))/8

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{\sqrt [3]{1+\frac{1}{x^2}}}{x^3} \, dx &=-\frac{3}{8} \left (1+\frac{1}{x^2}\right )^{4/3}\\ \end{align*}

Mathematica [A]  time = 0.0065492, size = 13, normalized size = 1. \[ -\frac{3}{8} \left (\frac{1}{x^2}+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^(-2))^(1/3)/x^3,x]

[Out]

(-3*(1 + x^(-2))^(4/3))/8

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Maple [B]  time = 0.002, size = 22, normalized size = 1.7 \begin{align*} -{\frac{3\,{x}^{2}+3}{8\,{x}^{2}}\sqrt [3]{{\frac{{x}^{2}+1}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8/x^2*(x^2+1)*((x^2+1)/x^2)^(1/3)

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Maxima [A]  time = 0.993679, size = 12, normalized size = 0.92 \begin{align*} -\frac{3}{8} \,{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*(1/x^2 + 1)^(4/3)

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Fricas [B]  time = 1.54863, size = 57, normalized size = 4.38 \begin{align*} -\frac{3 \,{\left (x^{2} + 1\right )} \left (\frac{x^{2} + 1}{x^{2}}\right )^{\frac{1}{3}}}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*(x^2 + 1)*((x^2 + 1)/x^2)^(1/3)/x^2

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Sympy [B]  time = 0.877217, size = 31, normalized size = 2.38 \begin{align*} - \frac{3 \sqrt [3]{1 + \frac{1}{x^{2}}}}{8} - \frac{3 \sqrt [3]{1 + \frac{1}{x^{2}}}}{8 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(1 + x**(-2))**(1/3)/8 - 3*(1 + x**(-2))**(1/3)/(8*x**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\frac{1}{x^{2}} + 1\right )}^{\frac{1}{3}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((1/x^2 + 1)^(1/3)/x^3, x)