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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.002908, antiderivative size = 26, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0135066, 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])

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 57, normalized size = 2.2 \begin{align*}{\frac{1}{2}\sqrt{-1+x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{-1+x}\sqrt{1+x}}-{\frac{1}{2}\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0338, size = 36, normalized size = 1.38 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} - 1} x - \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \end{align*}

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))

________________________________________________________________________________________

Fricas [A]  time = 1.55154, size = 95, normalized size = 3.65 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x + \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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 3.87031, size = 133, normalized size = 5.12 \begin{align*} \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} \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.79116, size = 39, normalized size = 1.5 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x + \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \end{align*}

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 + log(abs(-sqrt(x + 1) + sqrt(x - 1)))