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

Optimal. Leaf size=37 \[ \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\cosh ^{-1}\left (\sqrt {x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 37, 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 = {280, 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[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/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 280

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*
x)^(m + 1)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p)/(c*(m + 2*n*p + 1)), x] + Dist[(2*a1*a2*n*p)/(m + 2*n*p + 1), Int[
(c*x)^m*(a1 + b1*x^n)^(p - 1)*(a2 + b2*x^n)^(p - 1), x], x] /; FreeQ[{a1, b1, a2, b2, c, m}, x] && EqQ[a2*b1 +
a1*b2, 0] && IGtQ[2*n, 0] && GtQ[p, 0] && 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 {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\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 = 0.03, size = 72, normalized size = 1.95 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (\sqrt {x}-1\right )+2 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{\sqrt {\sqrt {x}-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 1.17, size = 268, normalized size = 7.24 \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[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/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 = 1.24 \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((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(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 [B]  time = 0.18, size = 57, normalized size = 1.54 \begin {gather*} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} {\left (\sqrt {x} - 2\right )} + 2 \, \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((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 72, normalized size = 1.95 \begin {gather*} -\frac {\sqrt {\left (\sqrt {x}-1\right ) \left (\sqrt {x}+1\right )}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )}{\sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}}+\sqrt {\sqrt {x}-1}\, \left (\sqrt {x}+1\right )^{\frac {3}{2}}-\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(3/2)-(x^(1/2)-1)^(1/2)*(x^(1/2)+1)^(1/2)-((x^(1/2)-1)*(x^(1/2)+1))^(1/2)/(x^(1/
2)+1)^(1/2)/(x^(1/2)-1)^(1/2)*ln(x^(1/2)+(x-1)^(1/2))

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maxima [A]  time = 0.49, size = 26, normalized size = 0.70 \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((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.07, size = 41, normalized size = 1.11 \begin {gather*} \sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}-\ln \left (\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}+\sqrt {x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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