3.9 \(\int \frac{1}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx\)

Optimal. Leaf size=84 \[ \sqrt{\sqrt{x^2+1}+x}+\frac{1}{\sqrt{\sqrt{x^2+1}+x}}-\frac{1}{2 \left (\sqrt{x^2+1}+x\right )}+\frac{1}{2} \log \left (\sqrt{x^2+1}+x\right )-2 \log \left (\sqrt{\sqrt{x^2+1}+x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0586596, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2117, 1821, 1620} \[ \sqrt{\sqrt{x^2+1}+x}+\frac{1}{\sqrt{\sqrt{x^2+1}+x}}-\frac{1}{2 \left (\sqrt{x^2+1}+x\right )}+\frac{1}{2} \log \left (\sqrt{x^2+1}+x\right )-2 \log \left (\sqrt{\sqrt{x^2+1}+x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]

[Out]

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

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 1821

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] -
1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && Intege
rQ[Simplify[(m + 1)/n]]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

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

Mathematica [A]  time = 0.0406454, size = 84, normalized size = 1. \[ \sqrt{\sqrt{x^2+1}+x}+\frac{1}{\sqrt{\sqrt{x^2+1}+x}}-\frac{1}{2 \left (\sqrt{x^2+1}+x\right )}+\frac{1}{2} \log \left (\sqrt{x^2+1}+x\right )-2 \log \left (\sqrt{\sqrt{x^2+1}+x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]

[Out]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \left ( 1+\sqrt{x+\sqrt{{x}^{2}+1}} \right ) ^{-1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + sqrt(x^2 + 1)) + 1), x)

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Fricas [A]  time = 1.07762, size = 197, normalized size = 2.35 \begin{align*} -\sqrt{x + \sqrt{x^{2} + 1}}{\left (x - \sqrt{x^{2} + 1} - 1\right )} + \frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x^{2} + 1} - 2 \, \log \left (\sqrt{x + \sqrt{x^{2} + 1}} + 1\right ) + \log \left (\sqrt{x + \sqrt{x^{2} + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x + sqrt(x^2 + 1)) + 1), x)