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

Optimal. Leaf size=12 \[ \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {619, 215} \[ \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + x + x^2],x]

[Out]

ArcSinh[(1 + 2*x)/Sqrt[3]]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + x + x^2],x]

[Out]

ArcSinh[(1 + 2*x)/Sqrt[3]]

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IntegrateAlgebraic [A]  time = 0.06, size = 20, normalized size = 1.67 \[ -\log \left (2 \sqrt {x^2+x+1}-2 x-1\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[1 + x + x^2],x]

[Out]

-Log[-1 - 2*x + 2*Sqrt[1 + x + x^2]]

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fricas [A]  time = 0.71, size = 18, normalized size = 1.50 \[ -\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-2*x + 2*sqrt(x^2 + x + 1) - 1)

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giac [A]  time = 0.65, size = 18, normalized size = 1.50 \[ -\log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-2*x + 2*sqrt(x^2 + x + 1) - 1)

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maple [A]  time = 0.35, size = 10, normalized size = 0.83

method	result	size
default	\(\arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )\)	\(10\)
trager	\(-\ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(2/3*(1/2+x)*3^(1/2))

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maxima [A]  time = 1.40, size = 11, normalized size = 0.92 \[ \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*sqrt(3)*(2*x + 1))

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mupad [B]  time = 0.24, size = 12, normalized size = 1.00 \[ \ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + x^2 + 1)^(1/2),x)

[Out]

log(x + (x + x^2 + 1)^(1/2) + 1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x**2 + x + 1), x)

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