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

Optimal. Leaf size=22 \[ \cosh ^{-1}(x)-\frac {\sqrt {x-1} \sqrt {x+1}}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {97, 52} \begin {gather*} \cosh ^{-1}(x)-\frac {\sqrt {x-1} \sqrt {x+1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x]*Sqrt[1 + x])/x^2,x]

[Out]

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

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rubi steps

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

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Mathematica [B]  time = 0.02, size = 52, normalized size = 2.36 \begin {gather*} \frac {-\sqrt {x+1} (x-1)-2 \sqrt {1-x} x \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{\sqrt {x-1} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 0.05, size = 48, normalized size = 2.18 \begin {gather*} 2 \tanh ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {x+1}}\right )-\frac {2 \sqrt {x-1}}{\sqrt {x+1} \left (\frac {x-1}{x+1}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 36, normalized size = 1.64 \begin {gather*} -\frac {x \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) + \sqrt {x + 1} \sqrt {x - 1} + x}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*log(sqrt(x + 1)*sqrt(x - 1) - x) + sqrt(x + 1)*sqrt(x - 1) + x)/x

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giac [B]  time = 1.32, size = 40, normalized size = 1.82 \begin {gather*} -\frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} - \frac {1}{2} \, \log \left ({\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 44, normalized size = 2.00 \begin {gather*} \frac {\sqrt {x -1}\, \sqrt {x +1}\, \left (x \ln \left (x +\sqrt {x^{2}-1}\right )-\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}-1}\, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.41, size = 27, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {x^{2} - 1}}{x} + \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.00, size = 109, normalized size = 4.95 \begin {gather*} 4\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )-\frac {\sqrt {x-1}-\mathrm {i}}{4\,\left (\sqrt {x+1}-1\right )}-\frac {\frac {5\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{4\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {1}{4}}{\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*atanh(((x - 1)^(1/2) - 1i)/((x + 1)^(1/2) - 1)) - ((x - 1)^(1/2) - 1i)/(4*((x + 1)^(1/2) - 1)) - ((5*((x - 1
)^(1/2) - 1i)^2)/(4*((x + 1)^(1/2) - 1)^2) + 1/4)/(((x - 1)^(1/2) - 1i)^3/((x + 1)^(1/2) - 1)^3 + ((x - 1)^(1/
2) - 1i)/((x + 1)^(1/2) - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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