3.11.67 \(\int \sqrt {1-x} \sqrt {1+x} \, dx\) [1067]

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

[Out]

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

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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, 222} \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 222

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*}

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Mathematica [A]
time = 0.04, size = 37, normalized size = 1.32 \begin {gather*} \frac {1}{2} x \sqrt {1-x^2}-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x]*Sqrt[1 + x],x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.42, size = 106, normalized size = 3.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-x \sqrt {1+x}+x^2 \sqrt {1+x}-2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}\right )}{2 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {\sqrt {1+x}}{\sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {5}{2}}}{2 \sqrt {1-x}}+\frac {3 \left (1+x\right )^{\frac {3}{2}}}{2 \sqrt {1-x}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(1 - x)^(1/2)*(1 + x)^(1/2),x]')

[Out]

Piecewise[{{I / 2 (-x Sqrt[1 + x] + x ^ 2 Sqrt[1 + x] - 2 ArcCosh[Sqrt[2] Sqrt[1 + x] / 2] Sqrt[-1 + x]) / Sqr
t[-1 + x], Abs[1 + x] > 2}}, ArcSin[Sqrt[2] Sqrt[1 + x] / 2] - Sqrt[1 + x] / Sqrt[1 - x] - (1 + x) ^ (5 / 2) /
 (2 Sqrt[1 - x]) + 3 (1 + x) ^ (3 / 2) / (2 Sqrt[1 - x])]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(20)=40\).
time = 0.16, size = 57, normalized size = 2.04

method result size
default \(\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) \(57\)
risch \(-\frac {x \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) \(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*arcsin
(x)

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Maxima [A]
time = 0.36, size = 17, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (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="maxima")

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Fricas [A]
time = 0.30, 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)

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Sympy [C] Result contains complex when optimal does not.
time = 1.61, size = 131, normalized size = 4.68 \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}\: \left |{x + 1}\right | > 2 \\\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), (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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).
time = 0.01, size = 103, normalized size = 3.68 \begin {gather*} 2 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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

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