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

Optimal result

Integrand size = 9, antiderivative size = 27 \[ \int \sqrt {3+x^2} \, dx=\frac {1}{2} x \sqrt {3+x^2}+\frac {3}{2} \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 221} \[ \int \sqrt {3+x^2} \, dx=\frac {3}{2} \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {x^2+3} x \]

[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*} \text {integral}& = \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} \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \sqrt {3+x^2} \, dx=\frac {1}{2} x \sqrt {3+x^2}-\frac {3}{2} \log \left (-x+\sqrt {3+x^2}\right ) \]

[In]

Integrate[Sqrt[3 + x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {x^{2}+3}}{2}\)	\(21\)
risch	\(\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {x^{2}+3}}{2}\)	\(21\)
trager	\(\frac {x \sqrt {x^{2}+3}}{2}+\frac {3 \ln \left (x +\sqrt {x^{2}+3}\right )}{2}\)	\(24\)
meijerg	\(-\frac {3 \left (-\frac {2 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {\frac {x^{2}}{3}+1}}{3}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}}{3}\right )\right )}{4 \sqrt {\pi }}\)	\(37\)
pseudoelliptic	\(\frac {x \sqrt {x^{2}+3}}{2}+\frac {3 \ln \left (\frac {x +\sqrt {x^{2}+3}}{x}\right )}{4}-\frac {3 \ln \left (\frac {\sqrt {x^{2}+3}-x}{x}\right )}{4}\)	\(46\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sqrt {3+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 3} x - \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 3}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sqrt {3+x^2} \, dx=\frac {x \sqrt {x^{2} + 3}}{2} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {3} x}{3} \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \sqrt {3+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 3} x + \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sqrt {3+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 3} x - \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 3}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \sqrt {3+x^2} \, dx=\frac {3\,\mathrm {asinh}\left (\frac {\sqrt {3}\,x}{3}\right )}{2}+\frac {x\,\sqrt {x^2+3}}{2} \]

[In]

int((x^2 + 3)^(1/2),x)

[Out]

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