3.1.67 \(\int \frac {1}{\sqrt {-1+2 x^2}} \, dx\) [67]

Optimal. Leaf size=25 \[ \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2}}\right )}{\sqrt {2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {223, 212} \begin {gather*} \frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {2 x^2-1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + 2*x^2],x]

[Out]

ArcTanh[(Sqrt[2]*x)/Sqrt[-1 + 2*x^2]]/Sqrt[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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {-1+2 x^2}}\right )\\ &=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2}}\right )}{\sqrt {2}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(25)=50\).
time = 0.02, size = 54, normalized size = 2.16 \begin {gather*} \frac {-\log \left (\sqrt {2}-\frac {2 x}{\sqrt {-1+2 x^2}}\right )+\log \left (\sqrt {2}+\frac {2 x}{\sqrt {-1+2 x^2}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + 2*x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 22, normalized size = 0.88

method	result	size
default	\(\frac {\ln \left (\sqrt {2}\, x +\sqrt {2 x^{2}-1}\right ) \sqrt {2}}{2}\)	\(22\)
pseudoelliptic	\(\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2 x^{2}-1}\, \sqrt {2}}{2 x}\right )}{2}\)	\(24\)
trager	\(-\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\mathit {RootOf}\left (\textit {\_Z}^{2}-2\right ) \sqrt {2 x^{2}-1}+2 x \right )}{2}\)	\(31\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (2 x^{2}-1\right )}\, \sqrt {2}\, \arcsin \left (\sqrt {2}\, x \right )}{2 \sqrt {\mathrm {signum}\left (2 x^{2}-1\right )}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.44, size = 24, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sqrt {2 \, x^{2} - 1} + 4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.57, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sqrt {2 \, x^{2} - 1} x + 4 \, x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 26, normalized size = 1.04 \begin {gather*} \frac {\sqrt {2} \log {\left (2 x + \sqrt {2} \sqrt {2 x^{2} - 1} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.49, size = 36, normalized size = 1.44 \begin {gather*} \frac {1}{2} \, \sqrt {2 \, x^{2} - 1} x + \frac {1}{4} \, \sqrt {2} \log \left ({\left | -\sqrt {2} x + \sqrt {2 \, x^{2} - 1} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2*x^2 - 1)*x + 1/4*sqrt(2)*log(abs(-sqrt(2)*x + sqrt(2*x^2 - 1)))

________________________________________________________________________________________

Mupad [B]
time = 0.13, size = 21, normalized size = 0.84 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\sqrt {2}\,x+\sqrt {2\,x^2-1}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2 - 1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 14, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {2}}{2 \sqrt {-2 x^{2}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________