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

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

[Out]

(Sqrt[-1 + x]*Sqrt[1 + x])/x

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Rubi [A]  time = 0.0019538, antiderivative size = 18, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

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*}

Mathematica [A]  time = 0.0055836, size = 18, normalized size = 1. \[ \frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

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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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{x}\sqrt{-1+x}\sqrt{1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.70386, size = 15, normalized size = 0.83 \begin{align*} \frac{\sqrt{x^{2} - 1}}{x} \end{align*}

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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Fricas [A]  time = 1.53116, size = 45, normalized size = 2.5 \begin{align*} \frac{\sqrt{x + 1} \sqrt{x - 1} + x}{x} \end{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [A]  time = 1.49032, size = 28, normalized size = 1.56 \begin{align*} \frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} \end{align*}

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)