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

Optimal. Leaf size=18 \[ \frac {\sqrt {x-1} \sqrt {x+1}}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {95} \begin {gather*} \frac {\sqrt {x-1} \sqrt {x+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

Rule 95

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*} \int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx &=\frac {\sqrt {-1+x} \sqrt {1+x}}{x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x-1} \sqrt {x+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

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IntegrateAlgebraic [A]  time = 0.04, size = 29, normalized size = 1.61 \begin {gather*} \frac {2 \sqrt {x-1}}{\sqrt {x+1} \left (\frac {x-1}{x+1}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.51, size = 17, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x + 1} \sqrt {x - 1} + x}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 1.23, size = 21, normalized size = 1.17 \begin {gather*} \frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x -1}\, \sqrt {x +1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 11, normalized size = 0.61 \begin {gather*} \frac {\sqrt {x^{2} - 1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 - 1)/x

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mupad [B]  time = 1.17, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x-1}\,\sqrt {x+1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x - 1)^(1/2)*(x + 1)^(1/2))/x

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sympy [C]  time = 8.23, size = 58, normalized size = 3.22 \begin {gather*} - \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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