3.976 \(\int \frac{1}{x \sqrt{-1+x^4}} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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Rubi [A]  time = 0.0068633, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 63, 203} \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-1 + x^4]),x]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

Rule 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0027186, size = 14, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{x^4-1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-1 + x^4]),x]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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Maple [A]  time = 0.008, size = 11, normalized size = 0.8 \begin{align*} -{\frac{1}{2}\arctan \left ({\frac{1}{\sqrt{{x}^{4}-1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^4-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.50476, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(sqrt(x^4 - 1))

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Fricas [A]  time = 1.50864, size = 36, normalized size = 2.57 \begin{align*} \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(sqrt(x^4 - 1))

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Sympy [A]  time = 1.35233, size = 24, normalized size = 1.71 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{for}\: \frac{1}{\left |{x^{4}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*acosh(x**(-2))/2, 1/Abs(x**4) > 1), (-asin(x**(-2))/2, True))

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Giac [A]  time = 1.18723, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, \arctan \left (\sqrt{x^{4} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(sqrt(x^4 - 1))