3.9.79 \(\int \frac {1}{x-\sqrt {1+2 x^2}} \, dx\) [879]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {6874, 266, 399, 221, 385, 212} \begin {gather*} -\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 ) \end {gather*}

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 212

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

Rule 221

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 266

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 385

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 399

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

Rule 6874

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

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 )+\text {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*}

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Mathematica [A]
time = 0.10, size = 68, normalized size = 1.70 \begin {gather*} \left (1+\sqrt {2}\right ) \log \left (2 \left (-1+\sqrt {2}\right ) x+\left (-2+\sqrt {2}\right ) \sqrt {1+2 x^2}\right )-\log \left (-2+\sqrt {2}-2 x^2+x \sqrt {2+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + Sqrt[2])*Log[2*(-1 + Sqrt[2])*x + (-2 + Sqrt[2])*Sqrt[1 + 2*x^2]] - Log[-2 + Sqrt[2] - 2*x^2 + x*Sqrt[2 +
4*x^2]]

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Maple [A]
time = 0.13, size = 33, normalized size = 0.82

method	result	size
default	\(\arctanh \left (\frac {x}{\sqrt {2 x^{2}+1}}\right )-\frac {\ln \left (x^{2}+1\right )}{2}-\arcsinh \left (\sqrt {2}\, x \right ) \sqrt {2}\)	\(33\)
trager	\(\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right ) \ln \left (\frac {x +\sqrt {2 x^{2}+1}}{x^{2}+1}\right )-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right ) \sqrt {2 x^{2}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right ) x^{2}+x \sqrt {2 x^{2}+1}+x^{2}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right )+1}{x^{2}+1}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right )+\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right )^{2} x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right ) \sqrt {2 x^{2}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right ) x^{2}+x \sqrt {2 x^{2}+1}+x^{2}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} -1\right )+1}{x^{2}+1}\right )\)	\(211\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (32) = 64\).
time = 0.34, size = 90, normalized size = 2.25 \begin {gather*} \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 {gather*}

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)

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Sympy [A]
time = 0.07, size = 27, normalized size = 0.68 \begin {gather*} - \log {\left (- x + \sqrt {2 x^{2} + 1} \right )} - \sqrt {2} \operatorname {asinh}{\left (\sqrt {2} x \right )} \end {gather*}

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (32) = 64\).
time = 3.65, size = 88, normalized size = 2.20 \begin {gather*} \sqrt {2} \log \left (-\sqrt {2} x + \sqrt {2 \, x^{2} + 1}\right ) + \frac {1}{2} \, \log \left ({\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} + 2 \, \sqrt {2} + 3\right ) - \frac {1}{2} \, \log \left ({\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} - 2 \, \sqrt {2} + 3\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

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

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Mupad [B]
time = 3.50, size = 57, normalized size = 1.42 \begin {gather*} -\ln \left (x-\mathrm {i}\right )-\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{2}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{2}-\sqrt {2}\,\mathrm {asinh}\left (\sqrt {2}\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - (2*x^2 + 1)^(1/2)),x)

[Out]

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

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