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

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ 2 \sqrt {x+1}-2 \log \left (\sqrt {x+1}+1\right ) \]

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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IntegrateAlgebraic [A]  time = 0.01, size = 22, normalized size = 1.00 \[ 2 \sqrt {x+1}-2 \log \left (\sqrt {x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + Sqrt[1 + x])^(-1),x]

[Out]

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

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fricas [A]  time = 1.11, size = 18, normalized size = 0.82 \[ 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]

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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giac [A]  time = 0.84, size = 18, normalized size = 0.82 \[ 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]

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)

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maple [A]  time = 0.08, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(-2 \ln \left (1+\sqrt {1+x}\right )+2 \sqrt {1+x}\)	\(19\)
trager	\(2 \sqrt {1+x}-\ln \left (2 \sqrt {1+x}+2+x \right )\)	\(22\)
default	\(2 \sqrt {1+x}+\ln \left (-1+\sqrt {1+x}\right )-\ln \left (1+\sqrt {1+x}\right )-\ln \relax (x )\)	\(31\)
meijerg	\(\frac {-4 \sqrt {\pi }+4 \sqrt {\pi }\, \sqrt {1+x}-4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+x}}{2}\right )}{2 \sqrt {\pi }}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+(1+x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.72, size = 18, normalized size = 0.82 \[ 2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} + 1\right ) \]

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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mupad [B]  time = 0.13, size = 18, normalized size = 0.82 \[ 2\,\sqrt {x+1}-2\,\ln \left (\sqrt {x+1}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x + 1)^(1/2) + 1),x)

[Out]

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

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sympy [A]  time = 0.14, size = 19, normalized size = 0.86 \[ 2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} + 1 \right )} \]

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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