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

Optimal. Leaf size=23 \[ \frac {1}{3} (x+1)^{3/2}-\frac {1}{3} (x-1)^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6689} \[ \frac {1}{3} (x+1)^{3/2}-\frac {1}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-1 + x)^(3/2)/3 + (1 + x)^(3/2)/3

Rule 6689

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+x}+\sqrt {1+x}} \, dx &=-\left (\frac {1}{2} \int \left (\sqrt {-1+x}-\sqrt {1+x}\right ) \, dx\right )\\ &=-\frac {1}{3} (-1+x)^{3/2}+\frac {1}{3} (1+x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \[ \frac {1}{3} (x+1)^{3/2}-\frac {1}{3} (x-1)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(-1 + x)^(3/2) + (1 + x)^(3/2)/3

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fricas [A]  time = 0.45, size = 15, normalized size = 0.65 \[ \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)

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giac [A]  time = 0.19, size = 15, normalized size = 0.65 \[ \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x + 1)^(3/2) - 1/3*(x - 1)^(3/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.70 \[ -\frac {\left (x -1\right )^{\frac {3}{2}}}{3}+\frac {\left (x +1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + 1} + \sqrt {x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.84, size = 15, normalized size = 0.65 \[ \frac {{\left (x+1\right )}^{3/2}}{3}-\frac {{\left (x-1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)^(3/2)/3 - (x - 1)^(3/2)/3

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sympy [B]  time = 0.41, size = 51, normalized size = 2.22 \[ \frac {4 x}{3 \sqrt {x - 1} + 3 \sqrt {x + 1}} + \frac {2 \sqrt {x - 1} \sqrt {x + 1}}{3 \sqrt {x - 1} + 3 \sqrt {x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x/(3*sqrt(x - 1) + 3*sqrt(x + 1)) + 2*sqrt(x - 1)*sqrt(x + 1)/(3*sqrt(x - 1) + 3*sqrt(x + 1))

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