3.9.34 \(\int \frac {1}{x \sqrt {a-b x^4}} \, dx\) [834]

Optimal. Leaf size=28 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {272, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a - b*x^4]),x]

[Out]

-1/2*ArcTanh[Sqrt[a - b*x^4]/Sqrt[a]]/Sqrt[a]

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 214

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

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a - b*x^4]),x]

[Out]

-1/2*ArcTanh[Sqrt[a - b*x^4]/Sqrt[a]]/Sqrt[a]

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Maple [A]
time = 0.14, size = 30, normalized size = 1.07

method	result	size
default	\(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}\)	\(30\)
elliptic	\(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2 \sqrt {a}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^(1/2)*ln((2*a+2*a^(1/2)*(-b*x^4+a)^(1/2))/x^2)

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Maxima [A]
time = 0.50, size = 39, normalized size = 1.39 \begin {gather*} \frac {\log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right )}{4 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 65, normalized size = 2.32 \begin {gather*} \left [\frac {\log \left (\frac {b x^{4} + 2 \, \sqrt {-b x^{4} + a} \sqrt {a} - 2 \, a}{x^{4}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{a}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*log((b*x^4 + 2*sqrt(-b*x^4 + a)*sqrt(a) - 2*a)/x^4)/sqrt(a), 1/2*sqrt(-a)*arctan(sqrt(-b*x^4 + a)*sqrt(-a
)/a)/a]

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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 53, normalized size = 1.89 \begin {gather*} \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2 \sqrt {a}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.62, size = 24, normalized size = 0.86 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.21, size = 20, normalized size = 0.71 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a - b*x^4)^(1/2)),x)

[Out]

-atanh((a - b*x^4)^(1/2)/a^(1/2))/(2*a^(1/2))

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