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

Optimal. Leaf size=35 \[ \frac{\tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.0430031, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1107, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a - b*x^2 + c*x^4),x]

[Out]

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

Rule 1107

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

Rule 618

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

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0087044, size = 41, normalized size = 1.17 \[ \frac{\tan ^{-1}\left (\frac{2 c x^2-b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a - b*x^2 + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.159, size = 38, normalized size = 1.1 \begin{align*}{\arctan \left ({(2\,c{x}^{2}-b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44559, size = 290, normalized size = 8.29 \begin{align*} \left [\frac{\log \left (\frac{2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} - b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c}}, -\frac{\sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} - b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.471686, size = 131, normalized size = 3.74 \begin{align*} - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} - b}{2 c} \right )}}{2} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} - b}{2 c} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.40388, size = 50, normalized size = 1.43 \begin{align*} \frac{\arctan \left (\frac{2 \, c x^{2} - b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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