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

Optimal. Leaf size=47 \[ -\frac{\tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.042422, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1114, 724, 204} \[ -\frac{\tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

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

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 204

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

Rubi steps

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

Mathematica [A]  time = 0.0142757, size = 46, normalized size = 0.98 \[ \frac{\tan ^{-1}\left (\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.165, size = 45, normalized size = 1. \begin{align*} -{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62648, size = 294, normalized size = 6.26 \begin{align*} \left [-\frac{\sqrt{-a} \log \left (\frac{{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{-a} + 8 \, a^{2}}{x^{4}}\right )}{4 \, a}, \frac{\arctan \left (\frac{\sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{a}}{2 \,{\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right )}{2 \, \sqrt{a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*sqrt(-a)*log(((b^2 - 4*a*c)*x^4 - 8*a*b*x^2 - 4*sqrt(c*x^4 + b*x^2 - a)*(b*x^2 - 2*a)*sqrt(-a) + 8*a^2)/
x^4)/a, 1/2*arctan(1/2*sqrt(c*x^4 + b*x^2 - a)*(b*x^2 - 2*a)*sqrt(a)/(a*c*x^4 + a*b*x^2 - a^2))/sqrt(a)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- a + b x^{2} + c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(-a + b*x**2 + c*x**4)), x)

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Giac [A]  time = 1.33649, size = 59, normalized size = 1.26 \begin{align*} \frac{\log \left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{2 \, \sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(-2*sqrt(-a)*(sqrt(c + b/x^2 - a/x^4) - sqrt(-a)/x^2) + b))/sqrt(-a)