3.2.19 \(\int \frac {1}{x^2 \sqrt {4+x^2}} \, dx\) [119]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {\sqrt {x^2+4}}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[4 + x^2]),x]

[Out]

-1/4*Sqrt[4 + x^2]/x

Rule 270

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*}

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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {4+x^2}}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[4 + x^2]),x]

[Out]

-1/4*Sqrt[4 + x^2]/x

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Mathics [A]
time = 2.08, size = 11, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {1+\frac {4}{x^2}}}{4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^2*Sqrt[4 + x^2]),x]')

[Out]

-Sqrt[1 + 4 / x ^ 2] / 4

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Maple [A]
time = 0.03, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
default	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
trager	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
risch	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
meijerg	\(-\frac {\sqrt {1+\frac {x^{2}}{4}}}{2 x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^2+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^{2} + 4}}{4 \, x} \end {gather*}

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 = 0.37, size = 14, normalized size = 0.88 \begin {gather*} -\frac {x + \sqrt {x^{2} + 4}}{4 \, x} \end {gather*}

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 = 0.41, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\sqrt {1 + \frac {4}{x^{2}}}}{4} \end {gather*}

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 = 0.00, size = 19, normalized size = 1.19 \begin {gather*} \frac {2}{\left (\sqrt {x^{2}+4}-x\right )^{2}-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((x - sqrt(x^2 + 4))^2 - 4)

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {x^2+4}}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x^2 + 4)^(1/2)),x)

[Out]

-(x^2 + 4)^(1/2)/(4*x)

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