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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0110565, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {402, 215, 377, 206} \[ \sinh ^{-1}(x)-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/(2 + x^2),x]

[Out]

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

Rule 402

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

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0126224, size = 27, normalized size = 1. \[ \sinh ^{-1}(x)-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2]/(2 + x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 23, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ( x \right ) -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{x\sqrt{2}}{2}{\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{x^{2} + 1} x}{x^{2} + 2} + \int \frac{\sqrt{x^{2} + 1} x^{4}}{x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1)*x/(x^2 + 2) + integrate(sqrt(x^2 + 1)*x^4/(x^6 + 5*x^4 + 8*x^2 + 4), x)

________________________________________________________________________________________

Fricas [B]  time = 1.7168, size = 173, normalized size = 6.41 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{9 \, x^{2} - 2 \, \sqrt{2}{\left (3 \, x^{2} + 2\right )} - 2 \, \sqrt{x^{2} + 1}{\left (3 \, \sqrt{2} x - 4 \, x\right )} + 6}{x^{2} + 2}\right ) - \log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log((9*x^2 - 2*sqrt(2)*(3*x^2 + 2) - 2*sqrt(x^2 + 1)*(3*sqrt(2)*x - 4*x) + 6)/(x^2 + 2)) - log(-x
+ sqrt(x^2 + 1))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{x^{2} + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**2 + 1)/(x**2 + 2), x)

________________________________________________________________________________________

Giac [B]  time = 1.07295, size = 86, normalized size = 3.19 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) - \log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*log(((x - sqrt(x^2 + 1))^2 - 2*sqrt(2) + 3)/((x - sqrt(x^2 + 1))^2 + 2*sqrt(2) + 3)) - log(-x + sq
rt(x^2 + 1))