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

Optimal. Leaf size=17 \[ 2 \sqrt{\sqrt{x^2+1}+1} \]

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

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Rubi [A]  time = 0.111079, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {6686} \[ 2 \sqrt{\sqrt{x^2+1}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{1+x^2} \sqrt{1+\sqrt{1+x^2}}} \, dx &=2 \sqrt{1+\sqrt{1+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0125079, size = 17, normalized size = 1. \[ 2 \sqrt{\sqrt{x^2+1}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

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Maple [A]  time = 0.007, size = 14, normalized size = 0.8 \begin{align*} 2\,\sqrt{\sqrt{{x}^{2}+1}+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.939651, size = 18, normalized size = 1.06 \begin{align*} 2 \, \sqrt{\sqrt{x^{2} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

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Fricas [A]  time = 0.479772, size = 36, normalized size = 2.12 \begin{align*} 2 \, \sqrt{\sqrt{x^{2} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

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Sympy [A]  time = 0.346681, size = 14, normalized size = 0.82 \begin{align*} 2 \sqrt{\sqrt{x^{2} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x**2 + 1) + 1)

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Giac [A]  time = 1.06909, size = 18, normalized size = 1.06 \begin{align*} 2 \, \sqrt{\sqrt{x^{2} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)