3.721 \(\int \frac{1}{\sqrt{1+x+\sqrt{-1+2 x}}} \, dx\)

Optimal. Leaf size=44 \[ 2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

[Out]

2*Sqrt[1 + x + Sqrt[-1 + 2*x]] - Sqrt[2]*ArcSinh[(1 + Sqrt[-1 + 2*x])/Sqrt[2]]

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Rubi [A]  time = 0.0351152, antiderivative size = 52, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {640, 619, 215} \[ \sqrt{2} \sqrt{2 x+2 \sqrt{2 x-1}+2}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x + Sqrt[-1 + 2*x]],x]

[Out]

Sqrt[2]*Sqrt[2 + 2*x + 2*Sqrt[-1 + 2*x]] - Sqrt[2]*ArcSinh[(1 + Sqrt[-1 + 2*x])/Sqrt[2]]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+x+\sqrt{-1+2 x}}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt{\frac{3}{2}+x+\frac{x^2}{2}}} \, dx,x,\sqrt{-1+2 x}\right )\\ &=\sqrt{2} \sqrt{2+2 x+2 \sqrt{-1+2 x}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{3}{2}+x+\frac{x^2}{2}}} \, dx,x,\sqrt{-1+2 x}\right )\\ &=\sqrt{2} \sqrt{2+2 x+2 \sqrt{-1+2 x}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{2}}} \, dx,x,1+\sqrt{-1+2 x}\right )\\ &=\sqrt{2} \sqrt{2+2 x+2 \sqrt{-1+2 x}}-\sqrt{2} \sinh ^{-1}\left (\frac{1+\sqrt{-1+2 x}}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0175189, size = 44, normalized size = 1. \[ 2 \sqrt{x+\sqrt{2 x-1}+1}-\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-1}+1}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x + Sqrt[-1 + 2*x]],x]

[Out]

2*Sqrt[1 + x + Sqrt[-1 + 2*x]] - Sqrt[2]*ArcSinh[(1 + Sqrt[-1 + 2*x])/Sqrt[2]]

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Maple [A]  time = 0.009, size = 38, normalized size = 0.9 \begin{align*} \sqrt{4\,x+4+4\,\sqrt{2\,x-1}}-{\it Arcsinh} \left ({\frac{\sqrt{2}}{2} \left ( 1+\sqrt{2\,x-1} \right ) } \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*x+4+4*(2*x-1)^(1/2))^(1/2)-arcsinh(1/2*(1+(2*x-1)^(1/2))*2^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + \sqrt{2 \, x - 1} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.8144, size = 246, normalized size = 5.59 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (-8 \, x^{2} - 8 \,{\left (2 \, x + 1\right )} \sqrt{2 \, x - 1} + 2 \,{\left (\sqrt{2}{\left (2 \, x + 3\right )} \sqrt{2 \, x - 1} + \sqrt{2}{\left (6 \, x - 1\right )}\right )} \sqrt{x + \sqrt{2 \, x - 1} + 1} - 24 \, x + 7\right ) + 2 \, \sqrt{x + \sqrt{2 \, x - 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(-8*x^2 - 8*(2*x + 1)*sqrt(2*x - 1) + 2*(sqrt(2)*(2*x + 3)*sqrt(2*x - 1) + sqrt(2)*(6*x - 1))*s
qrt(x + sqrt(2*x - 1) + 1) - 24*x + 7) + 2*sqrt(x + sqrt(2*x - 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x + \sqrt{2 x - 1} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x + sqrt(2*x - 1) + 1), x)

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Giac [A]  time = 1.22226, size = 92, normalized size = 2.09 \begin{align*} -\sqrt{2}{\left (\sqrt{3} + \log \left (\sqrt{3} - 1\right )\right )} + \sqrt{2} \log \left (\sqrt{2 \, x + 2 \, \sqrt{2 \, x - 1} + 2} - \sqrt{2 \, x - 1} - 1\right ) + \sqrt{2} \sqrt{2 \, x + 2 \, \sqrt{2 \, x - 1} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*(sqrt(3) + log(sqrt(3) - 1)) + sqrt(2)*log(sqrt(2*x + 2*sqrt(2*x - 1) + 2) - sqrt(2*x - 1) - 1) + sqr
t(2)*sqrt(2*x + 2*sqrt(2*x - 1) + 2)