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

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Rubi [A]  time = 0.13, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 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 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 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]]

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*}

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Mathematica [A]  time = 0.04, size = 42, normalized size = 1.00 \[ \frac {\sqrt {2 x^2+1}}{2 x}+x-\frac {1}{2 x}-\frac {\sinh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 44, normalized size = 1.05 \[ \frac {\sqrt {2} x \log \left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right ) + 2 \, x^{2} + \sqrt {2 \, x^{2} + 1} - 1}{2 \, x} \]

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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giac [A]  time = 0.46, size = 57, normalized size = 1.36 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 1}\right ) + x - \frac {\sqrt {2}}{{\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} - 1} - \frac {1}{2 \, x} \]

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

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maple [A]  time = 0.01, size = 45, normalized size = 1.07 \[ x -\sqrt {2 x^{2}+1}\, x -\frac {\sqrt {2}\, \arcsinh \left (\sqrt {2}\, x \right )}{2}-\frac {1}{2 x}+\frac {\left (2 x^{2}+1\right )^{\frac {3}{2}}}{2 x} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ x - \int \frac {1}{\sqrt {2 \, x^{2} + 1} + 1}\,{d x} \]

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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mupad [B]  time = 3.39, size = 31, normalized size = 0.74 \[ x-\frac {\sqrt {2}\,\mathrm {asinh}\left (\sqrt {2}\,x\right )}{2}+\frac {\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}-\frac {1}{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {2 x^{2} + 1}}{\sqrt {2 x^{2} + 1} + 1}\, dx \]

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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