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

Optimal. Leaf size=40 \[ -\frac{1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac{x}{\sqrt{2 x^2+1}}\right )-\sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0429, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 260, 402, 215, 377, 206} \[ -\frac{1}{2} \log \left (x^2+1\right )+\tanh ^{-1}\left (\frac{x}{\sqrt{2 x^2+1}}\right )-\sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x - Sqrt[1 + 2*x^2])^(-1),x]

[Out]

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

Rule 6742

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

Mathematica [A]  time = 0.0392163, size = 74, normalized size = 1.85 \[ \frac{1}{4} \left (-2 \log \left (x^2+1\right )-\log \left (3 x^2-2 \sqrt{2 x^2+1} x+1\right )+\log \left (3 x^2+2 \sqrt{2 x^2+1} x+1\right )-4 \sqrt{2} \sinh ^{-1}\left (\sqrt{2} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[2]*ArcSinh[Sqrt[2]*x] - 2*Log[1 + x^2] - Log[1 + 3*x^2 - 2*x*Sqrt[1 + 2*x^2]] + Log[1 + 3*x^2 + 2*x*S
qrt[1 + 2*x^2]])/4

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 33, normalized size = 0.8 \begin{align*}{\it Artanh} \left ({x{\frac{1}{\sqrt{2\,{x}^{2}+1}}}} \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-{\it Arcsinh} \left ( x\sqrt{2} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x - sqrt(2*x^2 + 1)), x)

________________________________________________________________________________________

Fricas [B]  time = 1.43387, size = 236, normalized size = 5.9 \begin{align*} \sqrt{2} \log \left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (\frac{2 \, x^{2} - \sqrt{2 \, x^{2} + 1}{\left (x + 1\right )} + x + 1}{x^{2}}\right ) + \frac{1}{2} \, \log \left (\frac{2 \, x^{2} + \sqrt{2 \, x^{2} + 1}{\left (x - 1\right )} - x + 1}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.196709, size = 27, normalized size = 0.68 \begin{align*} - \log{\left (x - \sqrt{2 x^{2} + 1} \right )} - \sqrt{2} \operatorname{asinh}{\left (\sqrt{2} x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - sqrt(2*x**2 + 1)) - sqrt(2)*asinh(sqrt(2)*x)

________________________________________________________________________________________

Giac [B]  time = 1.09438, size = 119, normalized size = 2.98 \begin{align*} \sqrt{2} \log \left (-\sqrt{2} x + \sqrt{2 \, x^{2} + 1}\right ) - \frac{1}{2} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (\frac{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} - 2 \, \sqrt{2} + 3}{{\left (\sqrt{2} x - \sqrt{2 \, x^{2} + 1}\right )}^{2} + 2 \, \sqrt{2} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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