3.119 $$\int \frac{1}{x^2 \sqrt{4+x^2}} \, dx$$

Optimal. Leaf size=16 $-\frac{\sqrt{x^2+4}}{4 x}$

[Out]

-Sqrt[4 + x^2]/(4*x)

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Rubi [A]  time = 0.0027984, 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} $-\frac{\sqrt{x^2+4}}{4 x}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^2*Sqrt[4 + x^2]),x]

[Out]

-Sqrt[4 + x^2]/(4*x)

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

Mathematica [A]  time = 0.0026909, size = 16, normalized size = 1. $-\frac{\sqrt{x^2+4}}{4 x}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^2*Sqrt[4 + x^2]),x]

[Out]

-Sqrt[4 + x^2]/(4*x)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,x}\sqrt{{x}^{2}+4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39813, size = 16, normalized size = 1. \begin{align*} -\frac{\sqrt{x^{2} + 4}}{4 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(x^2 + 4)/x

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Fricas [A]  time = 2.04558, size = 38, normalized size = 2.38 \begin{align*} -\frac{x + \sqrt{x^{2} + 4}}{4 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(x + sqrt(x^2 + 4))/x

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Sympy [A]  time = 0.711962, size = 12, normalized size = 0.75 \begin{align*} - \frac{\sqrt{1 + \frac{4}{x^{2}}}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + 4/x**2)/4

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Giac [A]  time = 1.05554, size = 26, normalized size = 1.62 \begin{align*} \frac{2}{{\left (x - \sqrt{x^{2} + 4}\right )}^{2} - 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((x - sqrt(x^2 + 4))^2 - 4)