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

Optimal. Leaf size=35 \[ \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\cosh ^{-1}\left (\sqrt {x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {323, 330, 52} \begin {gather*} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + ArcCosh[Sqrt[x]]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 323

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(2
*n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*(a2 + b2*x^n)^(p + 1))/(b1*b2*(m + 2*n*p + 1)), x] - Dist[(a
1*a2*c^(2*n)*(m - 2*n + 1))/(b1*b2*(m + 2*n*p + 1)), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x],
x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m +
2*n*p + 1, 0] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rule 330

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =
Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + (b1*x^(k*n))/c^n)^p*(a2 + (b2*x^(k*n))/c^n)^p, x]
, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio
nQ[m] && IntBinomialQ[a1*a2, b1*b2, c, 2*n, m, p, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 55, normalized size = 1.57 \begin {gather*} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}}{\sqrt {\sqrt {x}+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] + 2*ArcTanh[Sqrt[-1 + Sqrt[x]]/Sqrt[1 + Sqrt[x]]]

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IntegrateAlgebraic [B]  time = 1.27, size = 268, normalized size = 7.66 \begin {gather*} \frac {-4 \sqrt {\sqrt {x}+1} \left (-20 x^{3/2}-4 x+96 \sqrt {x}+48\right )-4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (-7 x^{3/2}-28 x+10 \sqrt {x}+84\right )+\sqrt {3} \left (-4 \sqrt {\sqrt {x}-1} \left (12 x^{3/2}+20 x-32 \sqrt {x}-48\right )-4 \left (14 x^{3/2}+4 x^2-18 x-70 \sqrt {x}-28\right )\right )}{\sqrt {3} \sqrt {\sqrt {x}+1} \left (-48 \sqrt {x}-32\right )+\sqrt {\sqrt {x}-1} \left (\sqrt {3} \sqrt {\sqrt {x}+1} \left (-16 \sqrt {x}-56\right )+80 \sqrt {x}+96\right )+28 x+112 \sqrt {x}+56}-4 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}-1}{\sqrt {3}-\sqrt {\sqrt {x}+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[1 + Sqrt[x]]*(48 + 96*Sqrt[x] - 4*x - 20*x^(3/2)) - 4*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(84 + 10*S
qrt[x] - 28*x - 7*x^(3/2)) + Sqrt[3]*(-4*Sqrt[-1 + Sqrt[x]]*(-48 - 32*Sqrt[x] + 20*x + 12*x^(3/2)) - 4*(-28 -
70*Sqrt[x] - 18*x + 14*x^(3/2) + 4*x^2)))/(56 + Sqrt[3]*(-32 - 48*Sqrt[x])*Sqrt[1 + Sqrt[x]] + Sqrt[-1 + Sqrt[
x]]*(96 + Sqrt[3]*(-56 - 16*Sqrt[x])*Sqrt[1 + Sqrt[x]] + 80*Sqrt[x]) + 112*Sqrt[x] + 28*x) - 4*ArcTanh[(-1 + S
qrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]

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fricas [A]  time = 0.41, size = 46, normalized size = 1.31 \begin {gather*} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {1}{2} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 39, normalized size = 1.11 \begin {gather*} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*log(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))

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maple [A]  time = 0.06, size = 41, normalized size = 1.17 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (\ln \left (\sqrt {x}+\sqrt {x -1}\right )+\sqrt {x -1}\, \sqrt {x}\right )}{\sqrt {x -1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 24, normalized size = 0.69 \begin {gather*} \sqrt {x - 1} \sqrt {x} + \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x - 1)*sqrt(x) + log(2*sqrt(x - 1) + 2*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 27.91, size = 83, normalized size = 2.37 \begin {gather*} \frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{x}} \right )}}{2 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x}} \right )}}{2 \pi ^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), 1/x)/(2*pi**(3/2)) - I*meijerg
(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), exp_polar(2*I*pi)/x)/(2*pi**(3/2))

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