\(\int \sqrt {1+x^2} \, dx\) [131]

Optimal result

Integrand size = 9, antiderivative size = 21 \[ \int \sqrt {1+x^2} \, dx=\frac {1}{2} x \sqrt {1+x^2}+\frac {\text {arcsinh}(x)}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, 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} \[ \int \sqrt {1+x^2} \, dx=\frac {\text {arcsinh}(x)}{2}+\frac {1}{2} \sqrt {x^2+1} x \]

[In]

Int[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/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 {1+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx \\ & = \frac {1}{2} x \sqrt {1+x^2}+\frac {\text {arcsinh}(x)}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \sqrt {1+x^2} \, dx=\frac {1}{2} x \sqrt {1+x^2}-\frac {1}{2} \log \left (-x+\sqrt {1+x^2}\right ) \]

[In]

Integrate[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 - Log[-x + Sqrt[1 + x^2]]/2

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {\operatorname {arcsinh}\left (x \right )}{2}+\frac {x \sqrt {x^{2}+1}}{2}\)	\(16\)
risch	\(\frac {\operatorname {arcsinh}\left (x \right )}{2}+\frac {x \sqrt {x^{2}+1}}{2}\)	\(16\)
trager	\(\frac {x \sqrt {x^{2}+1}}{2}+\frac {\ln \left (x +\sqrt {x^{2}+1}\right )}{2}\)	\(24\)
meijerg	\(-\frac {-2 \sqrt {\pi }\, x \sqrt {x^{2}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (x \right )}{4 \sqrt {\pi }}\)	\(27\)
pseudoelliptic	\(\frac {x \sqrt {x^{2}+1}}{2}+\frac {\ln \left (\frac {x +\sqrt {x^{2}+1}}{x}\right )}{4}-\frac {\ln \left (\frac {\sqrt {x^{2}+1}-x}{x}\right )}{4}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \sqrt {1+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt {1+x^2} \, dx=\frac {x \sqrt {x^{2} + 1}}{2} + \frac {\operatorname {asinh}{\left (x \right )}}{2} \]

[In]

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

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt {1+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \operatorname {arsinh}\left (x\right ) \]

[In]

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

[Out]

1/2*sqrt(x^2 + 1)*x + 1/2*arcsinh(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \sqrt {1+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sqrt {1+x^2} \, dx=\frac {\mathrm {asinh}\left (x\right )}{2}+\frac {x\,\sqrt {x^2+1}}{2} \]

[In]

int((x^2 + 1)^(1/2),x)

[Out]

asinh(x)/2 + (x*(x^2 + 1)^(1/2))/2