3.1008 \(\int \frac {1}{x \sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\)

Optimal. Leaf size=27 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {4, 266, 63, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4]),x]

[Out]

-ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]]/(2*Sqrt[a])

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]

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 208

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

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]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.68, size = 63, normalized size = 2.33 \[ \left [\frac {\log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 23, normalized size = 0.85 \[ \frac {\arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 29, normalized size = 1.07 \[ -\frac {\ln \left (\frac {2 a +2 \sqrt {c \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2 \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(c*x^4+a)^(1/2),x)

[Out]

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

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maxima [A]  time = 2.29, size = 37, normalized size = 1.37 \[ \frac {\log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{4 \, \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.55, size = 19, normalized size = 0.70 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{2\,\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + c*x^4)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.25, size = 22, normalized size = 0.81 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{2 \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-asinh(sqrt(a)/(sqrt(c)*x**2))/(2*sqrt(a))

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