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

Optimal. Leaf size=36 \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0038911, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {94, 92, 203} \[ \tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right )-\frac{\sqrt{x-1} \sqrt{x+1}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x]/(x^2*Sqrt[1 + x]),x]

[Out]

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

Rule 94

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{-1+x}}{x^2 \sqrt{1+x}} \, dx &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x} \sqrt{1+x}\right )\\ &=-\frac{\sqrt{-1+x} \sqrt{1+x}}{x}+\tan ^{-1}\left (\sqrt{-1+x} \sqrt{1+x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0179896, size = 50, normalized size = 1.39 \[ \frac{\sqrt{\frac{x-1}{x+1}} \left (-x^2+\sqrt{x^2-1} x \tan ^{-1}\left (\sqrt{x^2-1}\right )+1\right )}{(x-1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 43, normalized size = 1.2 \begin{align*}{\frac{1}{x} \left ( -\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) x-\sqrt{{x}^{2}-1} \right ) \sqrt{x-1}\sqrt{1+x}{\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49192, size = 27, normalized size = 0.75 \begin{align*} -\frac{\sqrt{x^{2} - 1}}{x} - \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x^2 - 1)/x - arcsin(1/abs(x))

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Fricas [A]  time = 1.69141, size = 101, normalized size = 2.81 \begin{align*} \frac{2 \, x \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - \sqrt{x + 1} \sqrt{x - 1} - x}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*arctan(sqrt(x + 1)*sqrt(x - 1) - x) - sqrt(x + 1)*sqrt(x - 1) - x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x - 1}}{x^{2} \sqrt{x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x - 1)/(x**2*sqrt(x + 1)), x)

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Giac [A]  time = 1.146, size = 57, normalized size = 1.58 \begin{align*} -\frac{8}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} + 4} - 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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