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

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {4+x^2}}{4 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^2+4}}{4 x} \]

[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*} \text {integral}& = -\frac {\sqrt {4+x^2}}{4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {4+x^2}}{4 x} \]

[In]

Integrate[1/(x^2*Sqrt[4 + x^2]),x]

[Out]

-1/4*Sqrt[4 + x^2]/x

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
pseudoelliptic	\(-\frac {\sqrt {x^{2}+4}}{4 x}\)	\(13\)
meijerg	\(-\frac {\sqrt {1+\frac {x^{2}}{4}}}{2 x}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {x + \sqrt {x^{2} + 4}}{4 \, x} \]

[In]

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

[Out]

-1/4*(x + sqrt(x^2 + 4))/x

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=- \frac {\sqrt {1 + \frac {4}{x^{2}}}}{4} \]

[In]

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

[Out]

-sqrt(1 + 4/x**2)/4

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^{2} + 4}}{4 \, x} \]

[In]

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

[Out]

-1/4*sqrt(x^2 + 4)/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=\frac {2}{{\left (x - \sqrt {x^{2} + 4}\right )}^{2} - 4} \]

[In]

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

[Out]

2/((x - sqrt(x^2 + 4))^2 - 4)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {4+x^2}} \, dx=-\frac {\sqrt {x^2+4}}{4\,x} \]

[In]

int(1/(x^2*(x^2 + 4)^(1/2)),x)

[Out]

-(x^2 + 4)^(1/2)/(4*x)