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

Optimal. Leaf size=42 \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.129511, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {6740, 6742, 277, 215} \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + 2*x^2]/(1 + Sqrt[1 + 2*x^2]),x]

[Out]

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

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0378985, size = 42, normalized size = 1. \[ \frac{\sqrt{2 x^2+1}}{2 x}+x-\frac{1}{2 x}-\frac{\sinh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + 2*x^2]/(1 + Sqrt[1 + 2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 45, normalized size = 1.1 \begin{align*} x-{\frac{1}{2\,x}}+{\frac{1}{2\,x} \left ( 2\,{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-x\sqrt{2\,{x}^{2}+1}-{\frac{{\it Arcsinh} \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - integrate(1/(sqrt(2*x^2 + 1) + 1), x)

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Fricas [A]  time = 1.67909, size = 111, normalized size = 2.64 \begin{align*} \frac{\sqrt{2} x \log \left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right ) + 2 \, x^{2} + \sqrt{2 \, x^{2} + 1} - 1}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13607, size = 77, normalized size = 1.83 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{2} x + \sqrt{2 \, x^{2} + 1}\right ) + x - \frac{\sqrt{2}}{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} - 1} - \frac{1}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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