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

Optimal. Leaf size=28 \[ \frac {1}{2} \sqrt {1-x} \sqrt {x+1} x+\frac {1}{2} \sin ^{-1}(x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {38, 41, 216} \begin {gather*} \frac {1}{2} \sqrt {1-x} \sqrt {x+1} x+\frac {1}{2} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]*Sqrt[1 + x],x]

[Out]

(Sqrt[1 - x]*x*Sqrt[1 + x])/2 + ArcSin[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 41

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{2} \left (\sqrt {1-x^2} x+\sin ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]*Sqrt[1 + x],x]

[Out]

(x*Sqrt[1 - x^2] + ArcSin[x])/2

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.06, size = 73, normalized size = 2.61 \begin {gather*} \frac {\frac {\sqrt {1-x}}{\sqrt {x+1}}-\frac {(1-x)^{3/2}}{(x+1)^{3/2}}}{\left (\frac {1-x}{x+1}+1\right )^2}-\tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - x]*Sqrt[1 + x],x]

[Out]

(-((1 - x)^(3/2)/(1 + x)^(3/2)) + Sqrt[1 - x]/Sqrt[1 + x])/(1 + (1 - x)/(1 + x))^2 - ArcTan[Sqrt[1 - x]/Sqrt[1
+ x]]

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 38, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} x \sqrt {-x + 1} - \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}

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)*x*sqrt(-x + 1) - arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

________________________________________________________________________________________

giac [B]  time = 1.04, size = 42, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}

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

________________________________________________________________________________________

maple [B]  time = 0.00, size = 57, normalized size = 2.04 \begin {gather*} \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}-\frac {\left (-x +1\right )^{\frac {3}{2}} \sqrt {x +1}}{2}+\frac {\sqrt {-x +1}\, \sqrt {x +1}}{2} \end {gather*}

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)^(3/2)*(x+1)^(1/2)+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)*a
rcsin(x)

________________________________________________________________________________________

maxima [A]  time = 2.98, size = 17, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}

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*arcsin(x)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 37, normalized size = 1.32 \begin {gather*} \frac {x\,\sqrt {1-x}\,\sqrt {x+1}}{2}-\frac {\ln \left (x-\sqrt {1-x}\,\sqrt {x+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/2)*(x + 1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 2.73, size = 133, normalized size = 4.75 \begin {gather*} \begin {cases} - i \operatorname {acosh}{\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 {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {\sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________