∫ √ _x __________∘ −1 + √_x ∘____

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

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Rubi [A]
time = 0.01, 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 = {329, 336, 54} \begin {gather*} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,

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[a1

*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 +

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

\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}+\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(35)=70\).
time = 1.16, size = 265, normalized size = 7.57 \begin {gather*} \frac {4 \left (4 \sqrt {1+\sqrt {x}} \left (-12-24 \sqrt {x}+x+5 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (-84-10 \sqrt {x}+28 x+7 x^{3/2}\right )+\sqrt {3} \left (28+70 \sqrt {x}+18 x-14 x^{3/2}-4 x^2-4 \sqrt {-1+\sqrt {x}} \left (-12-8 \sqrt {x}+5 x+3 x^{3/2}\right )\right )\right )}{56-16 \sqrt {3} \sqrt {1+\sqrt {x}} \left (2+3 \sqrt {x}\right )+\sqrt {-1+\sqrt {x}} \left (96-8 \sqrt {3} \sqrt {1+\sqrt {x}} \left (7+2 \sqrt {x}\right )+80 \sqrt {x}\right )+112 \sqrt {x}+28 x}-4 \tanh ^{-1}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \end {gather*}

(4*(4*Sqrt[1 + Sqrt[x]]*(-12 - 24*Sqrt[x] + x + 5*x^(3/2)) + Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(-84 - 10*Sq

rt[x] + 28*x + 7*x^(3/2)) + Sqrt[3]*(28 + 70*Sqrt[x] + 18*x - 14*x^(3/2) - 4*x^2 - 4*Sqrt[-1 + Sqrt[x]]*(-12 -

8*Sqrt[x] + 5*x + 3*x^(3/2)))))/(56 - 16*Sqrt[3]*Sqrt[1 + Sqrt[x]]*(2 + 3*Sqrt[x]) + Sqrt[-1 + Sqrt[x]]*(96 -

8*Sqrt[3]*Sqrt[1 + Sqrt[x]]*(7 + 2*Sqrt[x]) + 80*Sqrt[x]) + 112*Sqrt[x] + 28*x) - 4*ArcTanh[(-1 + Sqrt[-1 + S

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method	result	size

derivativedivides	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (\sqrt {x}\, \sqrt {x -1}+\ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{\sqrt {x -1}}\)	\(41\)
default	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (\sqrt {x}\, \sqrt {x -1}+\ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{\sqrt {x -1}}\)	\(41\)

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

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

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Maxima [A]
time = 0.29, 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*}

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

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Fricas [A]
time = 0.59, 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*}

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

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [A]
time = 1.91, 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*}

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

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

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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*}

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3.11.13 \(\int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\) [1013]