3.3.61 \(\int \sqrt {3+x^2} \, dx\) [261]

Optimal. Leaf size=27 \[ \frac {1}{2} x \sqrt {3+x^2}+\frac {3}{2} \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 221} \begin {gather*} \frac {1}{2} \sqrt {x^2+3} x+\frac {3}{2} \sinh ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + x^2],x]

[Out]

(x*Sqrt[3 + x^2])/2 + (3*ArcSinh[x/Sqrt[3]])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{2} x \sqrt {3+x^2}+\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {3+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 + x^2],x]

[Out]

(x*Sqrt[3 + x^2])/2 + (3*ArcTanh[x/Sqrt[3 + x^2]])/2

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Mathics [A]
time = 1.82, size = 20, normalized size = 0.74 \begin {gather*} \frac {x \sqrt {3+x^2}}{2}+\frac {3 \text {ArcSinh}\left [\frac {\sqrt {3} x}{3}\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sqrt[x^2+3],x]')

[Out]

x Sqrt[3 + x ^ 2] / 2 + 3 ArcSinh[Sqrt[3] x / 3] / 2

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Maple [A]
time = 0.05, size = 21, normalized size = 0.78

method	result	size
default	\(\frac {3 \arcsinh \left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {x^{2}+3}}{2}\)	\(21\)
risch	\(\frac {3 \arcsinh \left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {x^{2}+3}}{2}\)	\(21\)
trager	\(\frac {x \sqrt {x^{2}+3}}{2}-\frac {3 \ln \left (x -\sqrt {x^{2}+3}\right )}{2}\)	\(26\)
meijerg	\(-\frac {3 \left (-\frac {2 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {\frac {x^{2}}{3}+1}}{3}-2 \sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {3}}{3}\right )\right )}{4 \sqrt {\pi }}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 20, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 3} x + \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.31, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 3} x - \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(x^2 + 3)*x - 3/2*log(-x + sqrt(x^2 + 3))

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Sympy [A]
time = 0.08, size = 24, normalized size = 0.89 \begin {gather*} \frac {x \sqrt {x^{2} + 3}}{2} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {3} x}{3} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+3)**(1/2),x)

[Out]

x*sqrt(x**2 + 3)/2 + 3*asinh(sqrt(3)*x/3)/2

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Giac [A]
time = 0.00, size = 33, normalized size = 1.22 \begin {gather*} \frac {2}{4} x \sqrt {x^{2}+3}-\frac {3}{2} \ln \left (\sqrt {x^{2}+3}-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)^(1/2),x)

[Out]

1/2*sqrt(x^2 + 3)*x - 3/2*log(-x + sqrt(x^2 + 3))

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Mupad [B]
time = 0.05, size = 20, normalized size = 0.74 \begin {gather*} \frac {3\,\mathrm {asinh}\left (\frac {\sqrt {3}\,x}{3}\right )}{2}+\frac {x\,\sqrt {x^2+3}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 3)^(1/2),x)

[Out]

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

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