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

Optimal. Leaf size=36 \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]

[Out]

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

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Rubi [A]  time = 0.0203589, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {519, 41, 216} \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 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 \frac{1}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}} \sqrt{1+x}} \, dx &=\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}}}\\ &=\frac{\sqrt{1-x} \int \frac{1}{\sqrt{1-x^2}} \, dx}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}}}\\ &=\frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-1-\sqrt{x}} \sqrt{-1+\sqrt{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0163369, size = 36, normalized size = 1. \[ \frac{\sqrt{1-x} \sin ^{-1}(x)}{\sqrt{-\sqrt{x}-1} \sqrt{\sqrt{x}-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.601, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{-1-\sqrt{x}}}}{\frac{1}{\sqrt{-1+\sqrt{x}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.70318, size = 194, normalized size = 5.39 \begin{align*} -i \, \log \left (\frac{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1} + i \, x - 1}{x}\right ) + i \, \log \left (\frac{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1} - i \, x - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log((sqrt(x + 1)*sqrt(sqrt(x) - 1)*sqrt(-sqrt(x) - 1) + I*x - 1)/x) + I*log((sqrt(x + 1)*sqrt(sqrt(x) - 1)*
sqrt(-sqrt(x) - 1) - I*x - 1)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + 1} \sqrt{\sqrt{x} - 1} \sqrt{-\sqrt{x} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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