3.1.8 \(\int \frac {1}{x \sqrt {-1+x^4}} \, dx\) [8]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {272, 65, 209} \begin {gather*} \frac {1}{2} \arctan \left (\sqrt {x^4-1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-1 + x^4]),x]

[Out]

ArcTan[Sqrt[-1 + x^4]]/2

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 {gather*} \begin {aligned} \text {Integral} &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^4\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^4}\right )\\ &=\frac {1}{2} \arctan \left (\sqrt {-1+x^4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {1}{2} \arctan \left (\sqrt {-1+x^4}\right ) \end {gather*}

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 = 0.24, size = 11, normalized size = 0.79

method	result	size
default	\(-\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2}\)	\(11\)
elliptic	\(-\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2}\)	\(11\)
pseudoelliptic	\(-\frac {\arctan \left (\frac {1}{\sqrt {x^{4}-1}}\right )}{2}\)	\(11\)
trager	\(\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{4}-1}}{x^{2}}\right )}{2}\)	\(28\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{4}-1\right )}\, \left (\left (-2 \ln \left (2\right )+4 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{4}+1}}{2}\right )\right )}{4 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{4}-1\right )}}\)	\(61\)

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

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Maxima [A]
time = 0.48, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 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]

1/2*arctan(sqrt(x^4 - 1))

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Fricas [A]
time = 0.56, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 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]

1/2*arctan(sqrt(x^4 - 1))

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Sympy [C] Result contains complex when optimal does not.
time = 0.68, size = 24, normalized size = 1.71 \begin {gather*} \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 {gather*}

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 = 0.43, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, \arctan \left (\sqrt {x^{4} - 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]

1/2*arctan(sqrt(x^4 - 1))

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Mupad [B]
time = 0.19, size = 10, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {x^4-1}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^4 - 1)^(1/2)),x)

[Out]

atan((x^4 - 1)^(1/2))/2

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Chatgpt [F] Failed to verify
time = 1.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {\sqrt {x^{4}-1}}{x^{2}}\right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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