3.5.49 \(\int \frac {(-\sqrt {1-x}-\sqrt {1+x}) (\sqrt {1-x}+\sqrt {1+x})}{x^2} \, dx\) [449]

Optimal. Leaf size=26 \[ \frac {2}{x}+\frac {2 \sqrt {1-x^2}}{x}+2 \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6820, 6874, 283, 222} \begin {gather*} 2 \text {ArcSin}(x)+\frac {2 \sqrt {1-x^2}}{x}+\frac {2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.50 \begin {gather*} \frac {2 \left (1+\sqrt {1-x^2}+2 x \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.26, size = 50, normalized size = 1.92

method	result	size
default	\(\frac {2}{x}-\frac {2 \left (-\arcsin \left (x \right ) x -\sqrt {-x^{2}+1}\right ) \sqrt {1-x}\, \sqrt {1+x}}{x \sqrt {-x^{2}+1}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 24, normalized size = 0.92 \begin {gather*} \frac {2 \, \sqrt {-x^{2} + 1}}{x} + \frac {2}{x} + 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))*((1-x)^(1/2)+(1+x)^(1/2))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 44, normalized size = 1.69 \begin {gather*} -\frac {2 \, {\left (2 \, x \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {2}{x^{2}}\, dx - \int \frac {2 \sqrt {1 - x} \sqrt {x + 1}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(2/x**2, x) - Integral(2*sqrt(1 - x)*sqrt(x + 1)/x**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (24) = 48\).
time = 4.20, size = 149, normalized size = 5.73 \begin {gather*} 2 \, \pi + \frac {8 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4} + \frac {2}{x} + 4 \, \arctan \left (\frac {\sqrt {x + 1} {\left (\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*pi + 8*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x +
1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^2 - 4) + 2/x + 4*arctan(1/2*sqrt(x + 1)*((sqrt(2) - sq
rt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1)))

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Mupad [B]
time = 3.79, size = 118, normalized size = 4.54 \begin {gather*} \frac {\frac {5\,{\left (\sqrt {1-x}-1\right )}^2}{2\,{\left (\sqrt {x+1}-1\right )}^2}-\frac {1}{2}}{\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}-\frac {{\left (\sqrt {1-x}-1\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}}-8\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )+\frac {\sqrt {1-x}-1}{2\,\left (\sqrt {x+1}-1\right )}+\frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*((1 - x)^(1/2) - 1)^2)/(2*((x + 1)^(1/2) - 1)^2) - 1/2)/(((1 - x)^(1/2) - 1)/((x + 1)^(1/2) - 1) - ((1 - x
)^(1/2) - 1)^3/((x + 1)^(1/2) - 1)^3) - 8*atan(((1 - x)^(1/2) - 1)/((x + 1)^(1/2) - 1)) + ((1 - x)^(1/2) - 1)/
(2*((x + 1)^(1/2) - 1)) + 2/x

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