3.2.97 \(\int \frac {1}{x \sqrt {-1+x^2-x^4}} \, dx\) [197]

Optimal. Leaf size=30 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {2-x^2}{2 \sqrt {-1+x^2-x^4}}\right ) \]

[Out]

-1/2*arctan(1/2*(-x^2+2)/(-x^4+x^2-1)^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1128, 738, 210} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {2-x^2}{2 \sqrt {-x^4+x^2-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-1 + x^2 - x^4]),x]

[Out]

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

Rule 210

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1128

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {-1+x^2-x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {-1+x-x^2}} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-2+x^2}{\sqrt {-1+x^2-x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {-2+x^2}{2 \sqrt {-1+x^2-x^4}}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 27, normalized size = 0.90 \begin {gather*} -i \tanh ^{-1}\left (x^2+i \sqrt {-1+x^2-x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-1 + x^2 - x^4]),x]

[Out]

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

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Maple [A]
time = 0.14, size = 23, normalized size = 0.77

method	result	size
default	\(\frac {\arctan \left (\frac {x^{2}-2}{2 \sqrt {-x^{4}+x^{2}-1}}\right )}{2}\)	\(23\)
elliptic	\(\frac {\arctan \left (\frac {x^{2}-2}{2 \sqrt {-x^{4}+x^{2}-1}}\right )}{2}\)	\(23\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{4}+x^{2}-1}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-x^4+x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arctan(1/2*(x^2-2)/(-x^4+x^2-1)^(1/2))

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Maxima [C] Result contains complex when optimal does not.
time = 4.22, size = 17, normalized size = 0.57 \begin {gather*} -\frac {1}{2} i \, \operatorname {arsinh}\left (-\frac {1}{3} \, \sqrt {3} + \frac {2 \, \sqrt {3}}{3 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^4+x^2-1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.61, size = 55, normalized size = 1.83 \begin {gather*} \frac {1}{4} i \, \log \left (\frac {x^{2} + 2 i \, \sqrt {-x^{4} + x^{2} - 1} - 2}{2 \, x^{2}}\right ) - \frac {1}{4} i \, \log \left (\frac {x^{2} - 2 i \, \sqrt {-x^{4} + x^{2} - 1} - 2}{2 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^4+x^2-1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- x^{4} + x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x**4+x**2-1)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-x**4 + x**2 - 1)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^4+x^2-1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.54, size = 32, normalized size = 1.07 \begin {gather*} \frac {\ln \left (\frac {1}{x^2}\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (x^2-2+\sqrt {-x^4+x^2-1}\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 - x^4 - 1)^(1/2)),x)

[Out]

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

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