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

Optimal. Leaf size=21 \[ \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 21, 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 = {195, 215} \[ \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

Rule 195

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 0.86 \[ \frac {1}{2} \left (\sqrt {x^2+1} x+\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2] + ArcSinh[x])/2

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fricas [A]  time = 0.42, size = 25, normalized size = 1.19 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.37, size = 25, normalized size = 1.19 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} x - \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {\sqrt {x^{2}+1}\, x}{2}+\frac {\arcsinh \relax (x )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.95, size = 15, normalized size = 0.71 \[ \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \operatorname {arsinh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^2 + 1)*x + 1/2*arcsinh(x)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.71 \[ \frac {\mathrm {asinh}\relax (x)}{2}+\frac {x\,\sqrt {x^2+1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)^(1/2),x)

[Out]

asinh(x)/2 + (x*(x^2 + 1)^(1/2))/2

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sympy [A]  time = 0.19, size = 15, normalized size = 0.71 \[ \frac {x \sqrt {x^{2} + 1}}{2} + \frac {\operatorname {asinh}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

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