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

Optimal. Leaf size=73 \[ \left (1-\sqrt{x+1}\right )^2-4 \sqrt{1-\sqrt{x+1}}+2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{1-\sqrt{x+1}}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

[Out]

2*Sqrt[1 + x] - 4*Sqrt[1 - Sqrt[1 + x]] + (1 - Sqrt[1 + x])^2 + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[
5]])/Sqrt[5]

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Rubi [A]  time = 0.270541, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1628, 618, 206} \[ \left (1-\sqrt{x+1}\right )^2-4 \sqrt{1-\sqrt{x+1}}+2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{1-\sqrt{x+1}}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[x/(x + Sqrt[1 - Sqrt[1 + x]]),x]

[Out]

2*Sqrt[1 + x] - 4*Sqrt[1 - Sqrt[1 + x]] + (1 - Sqrt[1 + x])^2 + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[
5]])/Sqrt[5]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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{x}{x+\sqrt{1-\sqrt{1+x}}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \left (-1+x^2\right )}{-1+\sqrt{1-x}+x^2} \, dx,x,\sqrt{1+x}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^2 (1+x) \left (-2+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt{1-\sqrt{1+x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (-1-x+x^3-\frac{1}{-1+x+x^2}\right ) \, dx,x,\sqrt{1-\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}-4 \sqrt{1-\sqrt{1+x}}+\left (1-\sqrt{1+x}\right )^2-4 \operatorname{Subst}\left (\int \frac{1}{-1+x+x^2} \, dx,x,\sqrt{1-\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}-4 \sqrt{1-\sqrt{1+x}}+\left (1-\sqrt{1+x}\right )^2+8 \operatorname{Subst}\left (\int \frac{1}{5-x^2} \, dx,x,1+2 \sqrt{1-\sqrt{1+x}}\right )\\ &=2 \sqrt{1+x}-4 \sqrt{1-\sqrt{1+x}}+\left (1-\sqrt{1+x}\right )^2+\frac{8 \tanh ^{-1}\left (\frac{1+2 \sqrt{1-\sqrt{1+x}}}{\sqrt{5}}\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0745484, size = 52, normalized size = 0.71 \[ x-4 \sqrt{1-\sqrt{x+1}}+\frac{8 \tanh ^{-1}\left (\frac{2 \sqrt{1-\sqrt{x+1}}+1}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - 4*Sqrt[1 - Sqrt[1 + x]] + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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Maple [A]  time = 0.006, size = 60, normalized size = 0.8 \begin{align*} \left ( 1-\sqrt{1+x} \right ) ^{2}-2+2\,\sqrt{1+x}-4\,\sqrt{1-\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(x/(x+(1-(1+x)^(1/2))^(1/2)),x)

[Out]

(1-(1+x)^(1/2))^2-2+2*(1+x)^(1/2)-4*(1-(1+x)^(1/2))^(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 = 1.46052, size = 104, normalized size = 1.42 \begin{align*}{\left (\sqrt{x + 1} - 1\right )}^{2} - \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} - 4 \, \sqrt{-\sqrt{x + 1} + 1} - 2 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x + 1) - 1)^2 - 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) - 4*sqrt(-sqrt(x + 1) + 1) - 2

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Fricas [A]  time = 1.08725, size = 305, normalized size = 4.18 \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 ) + x - 4 \, \sqrt{-\sqrt{x + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(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)) + x - 4*sqrt(-sqrt(x + 1)
+ 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError