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

Optimal. Leaf size=22 \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

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Rubi [A]  time = 0.0091146, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 190, 43} \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[1 + x])^(-1),x]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0077022, size = 22, normalized size = 1. \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[1 + x])^(-1),x]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

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Maple [A]  time = 0.001, size = 31, normalized size = 1.4 \begin{align*} 2\,\sqrt{1+x}+\ln \left ( -1+\sqrt{1+x} \right ) -\ln \left ( 1+\sqrt{1+x} \right ) -\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.93172, size = 24, normalized size = 1.09 \begin{align*} 2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66271, size = 54, normalized size = 2.45 \begin{align*} 2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.111209, size = 19, normalized size = 0.86 \begin{align*} 2 \sqrt{x + 1} - 2 \log{\left (\sqrt{x + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05055, size = 24, normalized size = 1.09 \begin{align*} 2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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