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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1958, 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)/(1 + x)]/x^2,x]

[Out]

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

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 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 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])

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, 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)/(1 + x)]/x^2,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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fricas [A]  time = 0.44, size = 36, normalized size = 1.00 \[ \frac {2 \, x \arctan \left (\sqrt {\frac {x - 1}{x + 1}}\right ) - {\left (x + 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.41, size = 51, normalized size = 1.42 \[ -\frac {1}{2} \, {\left (\pi - 2\right )} \mathrm {sgn}\left (x + 1\right ) + 2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) \mathrm {sgn}\left (x + 1\right ) - \frac {2 \, \mathrm {sgn}\left (x + 1\right )}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(pi - 2)*sgn(x + 1) + 2*arctan(-x + sqrt(x^2 - 1))*sgn(x + 1) - 2*sgn(x + 1)/((x - sqrt(x^2 - 1))^2 + 1)

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maple [B]  time = 0.02, size = 59, normalized size = 1.64 \[ \frac {\sqrt {\frac {x -1}{x +1}}\, \left (x +1\right ) \left (-\sqrt {x^{2}-1}\, x^{2}-x \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )+\left (x^{2}-1\right )^{\frac {3}{2}}\right )}{\sqrt {\left (x -1\right ) \left (x +1\right )}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x-1)/(x+1))^(1/2)*(x+1)*((x^2-1)^(3/2)-x^2*(x^2-1)^(1/2)-x*arctan(1/(x^2-1)^(1/2)))/((x-1)*(x+1))^(1/2)/x

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maxima [A]  time = 1.95, size = 41, normalized size = 1.14 \[ -\frac {2 \, \sqrt {\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} + 1} + 2 \, \arctan \left (\sqrt {\frac {x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 41, normalized size = 1.14 \[ 2\,\mathrm {atan}\left (\sqrt {\frac {x-1}{x+1}}\right )-\frac {2\,\sqrt {\frac {x-1}{x+1}}}{\frac {x-1}{x+1}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x - 1)/(x + 1))^(1/2)/x^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {x - 1}{x + 1}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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