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

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

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Rubi [A]  time = 0.0178845, 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 \[ \frac{1}{2} \sqrt{x-1} x \sqrt{x+1}-\frac{1}{2} \cosh ^{-1}(x) \]

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

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Rubi in Sympy [A]  time = 3.12646, size = 20, normalized size = 0.77 \[ \frac{x \sqrt{x - 1} \sqrt{x + 1}}{2} - \frac{\operatorname{acosh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+x)**(1/2)*(1+x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00887505, size = 36, normalized size = 1.38 \[ \frac{1}{2} \sqrt{x-1} x \sqrt{x+1}-\sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + x]*Sqrt[1 + x],x]

[Out]

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

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Maple [B]  time = 0.004, size = 57, normalized size = 2.2 \[{\frac{1}{2}\sqrt{-1+x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{-1+x}\sqrt{1+x}}-{\frac{1}{2}\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{1+x}}}} \]

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))

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Maxima [A]  time = 1.34002, size = 36, normalized size = 1.38 \[ \frac{1}{2} \, \sqrt{x^{2} - 1} x - \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x - 1),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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Fricas [A]  time = 0.224017, size = 126, normalized size = 4.85 \[ \frac{2 \, x^{4} -{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1} - 2 \, x^{2} +{\left (2 \, \sqrt{x + 1} \sqrt{x - 1} x - 2 \, x^{2} + 1\right )} \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right )}{2 \,{\left (2 \, \sqrt{x + 1} \sqrt{x - 1} x - 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.1713, size = 133, normalized size = 5.12 \[ \begin{cases} - \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} - \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} + \frac{\sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x + 1}} + \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{- x + 1}} - \frac{i \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

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*as
in(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(5/2)/(2*sqrt(-x + 1)) + 3*I*(x + 1)**(3/
2)/(2*sqrt(-x + 1)) - I*sqrt(x + 1)/sqrt(-x + 1), True))

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GIAC/XCAS [A]  time = 0.2429, size = 39, normalized size = 1.5 \[ \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x +{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*sqrt(x - 1),x, algorithm="giac")

[Out]

1/2*sqrt(x + 1)*sqrt(x - 1)*x + ln(abs(-sqrt(x + 1) + sqrt(x - 1)))