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

Optimal. Leaf size=26 \[ \frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{2} \cosh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {38, 52} \[ \frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{2} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x]*Sqrt[1 + x],x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.96 \[ \frac {(x-1) \sqrt {x+1} x+2 \sqrt {1-x} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2 \sqrt {x-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x]*Sqrt[1 + x],x]

[Out]

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

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fricas [A]  time = 0.90, size = 32, normalized size = 1.23 \[ \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x + \frac {1}{2} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.27, size = 41, normalized size = 1.58 \[ \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} {\left (x - 2\right )} + \sqrt {x + 1} \sqrt {x - 1} + \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.00, size = 57, normalized size = 2.19 \[ -\frac {\sqrt {\left (x +1\right ) \left (x -1\right )}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{2 \sqrt {x -1}\, \sqrt {x +1}}+\frac {\sqrt {x -1}\, \left (x +1\right )^{\frac {3}{2}}}{2}-\frac {\sqrt {x -1}\, \sqrt {x +1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.62, size = 27, normalized size = 1.04 \[ \frac {1}{2} \, \sqrt {x^{2} - 1} x - \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.08, size = 30, normalized size = 1.15 \[ \frac {x\,\sqrt {x-1}\,\sqrt {x+1}}{2}-\frac {\ln \left (x+\sqrt {x-1}\,\sqrt {x+1}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)^(1/2)*(x + 1)^(1/2),x)

[Out]

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

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sympy [B]  time = 2.59, size = 133, normalized size = 5.12 \[ \begin {cases} - \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} - \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {\sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {i \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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