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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 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])

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 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]

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 = 0.01, size = 47, normalized size = 1.00 \[ 2 \sqrt {x+\sqrt {x+1}}-\tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \]

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]])]

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 47, normalized size = 1.00 \[ 2 \, \sqrt {x + \sqrt {x + 1}} + \frac {1}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.43, size = 33, normalized size = 0.70 \[ 2 \, \sqrt {x + \sqrt {x + 1}} + \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 32, normalized size = 0.68 \[ -\ln \left (\sqrt {x +1}+\frac {1}{2}+\sqrt {x +\sqrt {x +1}}\right )+2 \sqrt {x +\sqrt {x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + \sqrt {x + 1}}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x+\sqrt {x+1}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + (x + 1)^(1/2))^(1/2),x)

[Out]

int(1/(x + (x + 1)^(1/2))^(1/2), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________