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

Optimal. Leaf size=41 \[ 2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{\sqrt{x+1}+1}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

[Out]

2*Sqrt[1 + x] + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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Rubi [A]  time = 0.193243, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {800, 618, 206} \[ 2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{\sqrt{x+1}+1}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/(x + Sqrt[1 + Sqrt[1 + x]]),x]

[Out]

2*Sqrt[1 + x] + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], 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{\sqrt{1+x}}{x+\sqrt{1+\sqrt{1+x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2+\sqrt{1+x}} \, dx,x,\sqrt{1+x}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{(-1+x) (1+x)^2}{-1+x+x^2} \, dx,x,\sqrt{1+\sqrt{1+x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (x-\frac{1}{-1+x+x^2}\right ) \, dx,x,\sqrt{1+\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}-4 \operatorname{Subst}\left (\int \frac{1}{-1+x+x^2} \, dx,x,\sqrt{1+\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}+8 \operatorname{Subst}\left (\int \frac{1}{5-x^2} \, dx,x,1+2 \sqrt{1+\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}+\frac{8 \tanh ^{-1}\left (\frac{1+2 \sqrt{1+\sqrt{1+x}}}{\sqrt{5}}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.05696, size = 41, normalized size = 1. \[ 2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{\sqrt{x+1}+1}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + x] + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \begin{align*} 2+2\,\sqrt{1+x}+{\frac{8\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5} \left ( 1+2\,\sqrt{1+\sqrt{1+x}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2+2*(1+x)^(1/2)+8/5*arctanh(1/5*(1+2*(1+(1+x)^(1/2))^(1/2))*5^(1/2))*5^(1/2)

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Maxima [A]  time = 1.42723, size = 69, normalized size = 1.68 \begin{align*} -\frac{4}{5} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - 2 \, \sqrt{\sqrt{x + 1} + 1} - 1}{\sqrt{5} + 2 \, \sqrt{\sqrt{x + 1} + 1} + 1}\right ) + 2 \, \sqrt{x + 1} + 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.00712, size = 284, normalized size = 6.93 \begin{align*} \frac{4}{5} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5}{\left (3 \, x + 1\right )} -{\left (\sqrt{5}{\left (x + 2\right )} - 5 \, x\right )} \sqrt{x + 1} +{\left (\sqrt{5}{\left (x + 2\right )} +{\left (\sqrt{5}{\left (2 \, x - 1\right )} - 5\right )} \sqrt{x + 1} - 5 \, x\right )} \sqrt{\sqrt{x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*sqrt(5)*log((2*x^2 - sqrt(5)*(3*x + 1) - (sqrt(5)*(x + 2) - 5*x)*sqrt(x + 1) + (sqrt(5)*(x + 2) + (sqrt(5)
*(2*x - 1) - 5)*sqrt(x + 1) - 5*x)*sqrt(sqrt(x + 1) + 1) + 3*x + 3)/(x^2 - x - 1)) + 2*sqrt(x + 1)

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Sympy [A]  time = 10.8422, size = 112, normalized size = 2.73 \begin{align*} 2 \sqrt{x + 1} - 16 \left (\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left (\frac{2 \sqrt{5} \left (\sqrt{\sqrt{x + 1} + 1} + \frac{1}{2}\right )}{5} \right )}}{10} & \text{for}\: \left (\sqrt{\sqrt{x + 1} + 1} + \frac{1}{2}\right )^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{2 \sqrt{5} \left (\sqrt{\sqrt{x + 1} + 1} + \frac{1}{2}\right )}{5} \right )}}{10} & \text{for}\: \left (\sqrt{\sqrt{x + 1} + 1} + \frac{1}{2}\right )^{2} < \frac{5}{4} \end{cases}\right ) + 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 16*Piecewise((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1)
+ 1) + 1/2)**2 > 5/4), (-sqrt(5)*atanh(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1) + 1) +
1/2)**2 < 5/4)) + 2

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

[Out]

Exception raised: NotImplementedError