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

Optimal. Leaf size=47 \[ 2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left (\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0295322, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {640, 621, 206} \[ 2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left (\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x + Sqrt[1 + x]],x]

[Out]

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

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 621

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x+\sqrt{1+x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x+x^2}} \, dx,x,\sqrt{1+x}\right )\\ &=2 \sqrt{x+\sqrt{1+x}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x+x^2}} \, dx,x,\sqrt{1+x}\right )\\ &=2 \sqrt{x+\sqrt{1+x}}-2 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{1+2 \sqrt{1+x}}{\sqrt{x+\sqrt{1+x}}}\right )\\ &=2 \sqrt{x+\sqrt{1+x}}-\tanh ^{-1}\left (\frac{1+2 \sqrt{1+x}}{2 \sqrt{x+\sqrt{1+x}}}\right )\\ \end{align*}

Mathematica [A]  time = 0.012928, size = 47, normalized size = 1. \[ 2 \sqrt{x+\sqrt{x+1}}-\tanh ^{-1}\left (\frac{2 \sqrt{x+1}+1}{2 \sqrt{x+\sqrt{x+1}}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x + Sqrt[1 + x]],x]

[Out]

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

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Maple [A]  time = 0.009, size = 32, normalized size = 0.7 \begin{align*} 2\,\sqrt{x+\sqrt{1+x}}-\ln \left ( \sqrt{1+x}+{\frac{1}{2}}+\sqrt{x+\sqrt{1+x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(x + sqrt(x + 1)), x)

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Fricas [A]  time = 3.67419, size = 143, normalized size = 3.04 \begin{align*} 2 \, \sqrt{x + \sqrt{x + 1}} + \frac{1}{2} \, \log \left (4 \, \sqrt{x + \sqrt{x + 1}}{\left (2 \, \sqrt{x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt{x + 1} - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + sqrt(x + 1)) + 1/2*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) - 8*x - 8*sqrt(x + 1) - 5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x + sqrt(x + 1)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError