3.9.77 \(\int \frac {1}{x \sqrt {1-x^4}} \, dx\) [877]

Optimal. Leaf size=16 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 212} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 - x^4]),x]

[Out]

-1/2*ArcTanh[Sqrt[1 - x^4]]

Rule 65

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 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 272

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]]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1-x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 - x^4]),x]

[Out]

-1/2*ArcTanh[Sqrt[1 - x^4]]

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Maple [A]
time = 0.21, size = 13, normalized size = 0.81

method	result	size
default	\(-\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}\)	\(13\)
elliptic	\(-\frac {\arctanh \left (\frac {1}{\sqrt {-x^{4}+1}}\right )}{2}\)	\(13\)
trager	\(-\frac {\ln \left (\frac {1+\sqrt {-x^{4}+1}}{x^{2}}\right )}{2}\)	\(19\)
meijerg	\(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )+\left (-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{4 \sqrt {\pi }}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 0.30, size = 29, normalized size = 1.81 \begin {gather*} -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 0.35, size = 29, normalized size = 1.81 \begin {gather*} -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.42, size = 24, normalized size = 1.50 \begin {gather*} \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {1}{x^{2}} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
time = 1.23, size = 31, normalized size = 1.94 \begin {gather*} -\frac {1}{4} \, \log \left (\sqrt {-x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (-\sqrt {-x^{4} + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.31, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\mathrm {atanh}\left (\sqrt {1-x^4}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1 - x^4)^(1/2)),x)

[Out]

-atanh((1 - x^4)^(1/2))/2

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