\(\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx\) [837]

Optimal result

Integrand size = 16, antiderivative size = 15 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {-1+x} \sqrt {1+x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {75} \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {x-1} \sqrt {x+1} \]

[In]

Int[x/(Sqrt[-1 + x]*Sqrt[1 + x]),x]

[Out]

Sqrt[-1 + x]*Sqrt[1 + x]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {-1+x} \sqrt {1+x} \]

[In]

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

[Out]

Sqrt[-1 + x]*Sqrt[1 + x]

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

method	result	size
gosper	\(\sqrt {-1+x}\, \sqrt {1+x}\)	\(12\)
default	\(\sqrt {-1+x}\, \sqrt {1+x}\)	\(12\)
risch	\(\sqrt {-1+x}\, \sqrt {1+x}\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {x + 1} \sqrt {x - 1} \]

[In]

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

[Out]

sqrt(x + 1)*sqrt(x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.57 (sec) , antiderivative size = 76, normalized size of antiderivative = 5.07 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 2.

Time = 0.21 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.47 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {x^{2} - 1} \]

[In]

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

[Out]

sqrt(x^2 - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=\sqrt {x + 1} \sqrt {x - 1} \]

[In]

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

[Out]

sqrt(x + 1)*sqrt(x - 1)

Mupad [B] (verification not implemented)

Time = 2.67 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.47 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx=-\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {x+1}-1\right )}^2\,\left (1+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {2\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}\right )} \]

[In]

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

[Out]

-(((x - 1)^(1/2) - 1i)^2*8i)/(((x + 1)^(1/2) - 1)^2*(((x - 1)^(1/2) - 1i)^4/((x + 1)^(1/2) - 1)^4 - (2*((x - 1
)^(1/2) - 1i)^2)/((x + 1)^(1/2) - 1)^2 + 1))