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

Optimal. Leaf size=9 \[ 2 \sqrt{x+1} \]

[Out]

2*Sqrt[1 + x]

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Rubi [A]  time = 0.0007305, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ 2 \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x],x]

[Out]

2*Sqrt[1 + x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0018038, size = 9, normalized size = 1. \[ 2 \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x],x]

[Out]

2*Sqrt[1 + x]

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Maple [A]  time = 0.001, size = 8, normalized size = 0.9 \begin{align*} 2\,\sqrt{1+x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.57436, size = 9, normalized size = 1. \begin{align*} 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Fricas [A]  time = 1.44447, size = 20, normalized size = 2.22 \begin{align*} 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Sympy [A]  time = 0.051254, size = 7, normalized size = 0.78 \begin{align*} 2 \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Giac [A]  time = 1.13343, size = 9, normalized size = 1. \begin{align*} 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)