3.452 \(\int \frac{-\sqrt{-1+x}+\sqrt{1+x}}{\sqrt{-1+x}+\sqrt{1+x}} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.142982, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2104, 6742, 38, 52} \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2104

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

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]

Rubi steps

\begin{align*} \int \frac{-\sqrt{-1+x}+\sqrt{1+x}}{\sqrt{-1+x}+\sqrt{1+x}} \, dx &=-\left (\frac{1}{2} \int \sqrt{-1+x} \left (-\sqrt{-1+x}+\sqrt{1+x}\right ) \, dx\right )+\frac{1}{2} \int \sqrt{1+x} \left (-\sqrt{-1+x}+\sqrt{1+x}\right ) \, dx\\ &=\frac{1}{2} \int \left (1+x-\sqrt{-1+x} \sqrt{1+x}\right ) \, dx-\frac{1}{2} \int \left (1-x+\sqrt{-1+x} \sqrt{1+x}\right ) \, dx\\ &=\frac{x^2}{2}-2 \left (\frac{1}{2} \int \sqrt{-1+x} \sqrt{1+x} \, dx\right )\\ &=\frac{x^2}{2}-2 \left (\frac{1}{4} \sqrt{-1+x} x \sqrt{1+x}-\frac{1}{4} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx\right )\\ &=\frac{x^2}{2}-2 \left (\frac{1}{4} \sqrt{-1+x} x \sqrt{1+x}-\frac{1}{4} \cosh ^{-1}(x)\right )\\ \end{align*}

Mathematica [A]  time = 0.161114, size = 58, normalized size = 1.76 \[ \frac{1}{2} \left (x^2-\sqrt{x-1} \sqrt{x+1} x+\frac{2 (x-1) \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )}{\sqrt{-(x-1)^2}}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 62, normalized size = 1.9 \begin{align*} -{\frac{1}{2}\sqrt{x-1} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{x-1}\sqrt{1+x}}+{\frac{1}{2}\sqrt{ \left ( x-1 \right ) \left ( 1+x \right ) }\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{x-1}}}{\frac{1}{\sqrt{1+x}}}}+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((sqrt(x + 1) - sqrt(x - 1))/(sqrt(x + 1) + sqrt(x - 1)), x)

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Fricas [A]  time = 0.958409, size = 109, normalized size = 3.3 \begin{align*} -\frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x + \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 25.2528, size = 226, normalized size = 6.85 \begin{align*} - \frac{\left (x - 1\right )^{\frac{5}{2}}}{4 \sqrt{x + 1}} - \frac{3 \left (x - 1\right )^{\frac{3}{2}}}{4 \sqrt{x + 1}} - \frac{\sqrt{x - 1}}{2 \sqrt{x + 1}} + \frac{\left (x - 1\right )^{2}}{4} + 2 \left (\begin{cases} \frac{\left (x + 1\right )^{2}}{8} + \frac{\operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{8 \sqrt{x - 1}} + \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{8 \sqrt{x - 1}} - \frac{\sqrt{x + 1}}{4 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{\left (x + 1\right )^{2}}{8} - \frac{i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{4} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{8 \sqrt{1 - x}} - \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{8 \sqrt{1 - x}} + \frac{i \sqrt{x + 1}}{4 \sqrt{1 - x}} & \text{otherwise} \end{cases}\right ) + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x - 1)**(5/2)/(4*sqrt(x + 1)) - 3*(x - 1)**(3/2)/(4*sqrt(x + 1)) - sqrt(x - 1)/(2*sqrt(x + 1)) + (x - 1)**2/
4 + 2*Piecewise(((x + 1)**2/8 + acosh(sqrt(2)*sqrt(x + 1)/2)/4 - (x + 1)**(5/2)/(8*sqrt(x - 1)) + 3*(x + 1)**(
3/2)/(8*sqrt(x - 1)) - sqrt(x + 1)/(4*sqrt(x - 1)), Abs(x + 1)/2 > 1), ((x + 1)**2/8 - I*asin(sqrt(2)*sqrt(x +
1)/2)/4 + I*(x + 1)**(5/2)/(8*sqrt(1 - x)) - 3*I*(x + 1)**(3/2)/(8*sqrt(1 - x)) + I*sqrt(x + 1)/(4*sqrt(1 - x
)), True)) + asinh(sqrt(2)*sqrt(x - 1)/2)/2

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Giac [A]  time = 1.1937, size = 57, normalized size = 1.73 \begin{align*} \frac{1}{2} \,{\left (x + 1\right )}^{2} - \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x - x - \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) - 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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