3.4.80 \(\int \frac {1}{\sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}} \sqrt {1+x}} \, dx\) [380]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {533, 41, 222} \begin {gather*} \frac {\sqrt {1-x} \text {ArcSin}(x)}{\sqrt {-\sqrt {x}-1} \sqrt {\sqrt {x}-1}} \end {gather*}

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 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]

Rule 533

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)^FracPa
rt[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])

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.18, size = 44, normalized size = 1.22 \begin {gather*} -i \log \left (-x+i \sqrt {-1-\sqrt {x}} \sqrt {-1+\sqrt {x}} \sqrt {1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-I)*Log[-x + I*Sqrt[-1 - Sqrt[x]]*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + x]]

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {x +1}\, \sqrt {-1-\sqrt {x}}\, \sqrt {-1+\sqrt {x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Result contains complex when optimal does not.
time = 1.90, size = 69, normalized size = 1.92 \begin {gather*} -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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \sqrt {x} - 1} \sqrt {\sqrt {x} - 1} \sqrt {x + 1}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\sqrt {x}-1}\,\sqrt {-\sqrt {x}-1}\,\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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