\(\int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx\) [840]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {-1+x} \sqrt {1+x}}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {97} \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {x-1} \sqrt {x+1}}{x} \]

[In]

Int[1/(Sqrt[-1 + x]*x^2*Sqrt[1 + x]),x]

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x} \sqrt {1+x}}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {-1+x} \sqrt {1+x}}{x} \]

[In]

Integrate[1/(Sqrt[-1 + x]*x^2*Sqrt[1 + x]),x]

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
gosper	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}}{x}\)	\(15\)
default	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}}{x}\)	\(15\)
risch	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}}{x}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {x + 1} \sqrt {x - 1} + x}{x} \]

[In]

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

[Out]

(sqrt(x + 1)*sqrt(x - 1) + x)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]

[In]

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

[Out]

-meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), x**(-2))/(4*pi**(3/2)) - I*meijerg(((1
/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {x^{2} - 1}}{x} \]

[In]

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

[Out]

sqrt(x^2 - 1)/x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} \]

[In]

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

[Out]

8/((sqrt(x + 1) - sqrt(x - 1))^4 + 4)

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {\sqrt {x-1}\,\sqrt {x+1}}{x} \]

[In]

int(1/(x^2*(x - 1)^(1/2)*(x + 1)^(1/2)),x)

[Out]

((x - 1)^(1/2)*(x + 1)^(1/2))/x