3.5.46 \(\int x (-\sqrt {1-x}-\sqrt {1+x}) (\sqrt {1-x}+\sqrt {1+x}) \, dx\) [446]

Optimal. Leaf size=21 \[ -x^2+\frac {2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6820, 6874, 267} \begin {gather*} \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-x^2 + (2*(1 - x^2)^(3/2))/3

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\int x \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx\\ &=-\int \left (2 x+2 x \sqrt {1-x^2}\right ) \, dx\\ &=-x^2-2 \int x \sqrt {1-x^2} \, dx\\ &=-x^2+\frac {2}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.19 \begin {gather*} -\frac {1}{3} (-1+x) (1+x) \left (3+2 \sqrt {1-x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

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

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Maple [A]
time = 0.25, size = 26, normalized size = 1.24

method	result	size
default	\(-x^{2}-\frac {2 \sqrt {1-x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 17, normalized size = 0.81 \begin {gather*} -x^{2} + \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 2/3*(-x^2 + 1)^(3/2)

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Fricas [A]
time = 0.35, size = 25, normalized size = 1.19 \begin {gather*} -x^{2} - \frac {2}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 54.64, size = 144, normalized size = 6.86 \begin {gather*} \frac {x^{3}}{3} + x - \frac {\left (x + 1\right )^{3}}{3} + 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 4 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) + 1 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + x - (x + 1)**3/3 + 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (sqrt(x
+ 1) < sqrt(2)) & (sqrt(x + 1) > -sqrt(2)))) - 4*Piecewise((x*sqrt(1 - x)*sqrt(x + 1)/4 - (1 - x)**(3/2)*(x +
1)**(3/2)/6 + asin(sqrt(2)*sqrt(x + 1)/2)/2, (sqrt(x + 1) < sqrt(2)) & (sqrt(x + 1) > -sqrt(2)))) + 1

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (17) = 34\).
time = 3.04, size = 54, normalized size = 2.57 \begin {gather*} -{\left (x + 1\right )}^{2} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + 2 \, x + 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)^2 - 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - sqrt(x + 1)*(x - 2)*sqrt(-x + 1) + 2*x + 2

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Mupad [B]
time = 3.06, size = 25, normalized size = 1.19 \begin {gather*} -x^2-\frac {2\,\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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