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 = 0.0752139, 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 \[ \frac{\tanh ^{-1}\left (\frac{b-2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

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]

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Rubi in Sympy [A]  time = 9.64212, size = 34, normalized size = 0.97 \[ - \frac{\operatorname{atanh}{\left (\frac{- b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(c*x**4-b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.014396, 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 = 0.002, size = 38, normalized size = 1.1 \[{1\arctan \left ({(2\,c{x}^{2}-b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.264619, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{b^{3} - 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c}}, \frac{\arctan \left (-\frac{{\left (2 \, c x^{2} - b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c}}\right ] \]

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((b^3 - 4*a*b*c - 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 - 2*b*c*x^2 + b^2
- 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 - b*x^2 + a))/sqrt(b^2 - 4*a*c), arctan(-(2*
c*x^2 - b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c))/sqrt(-b^2 + 4*a*c)]

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Sympy [A]  time = 1.33128, size = 131, normalized size = 3.74 \[ - \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{- 4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} + b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} - b}{2 c} \right )}}{2} + \frac{\sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x^{2} + \frac{4 a c \sqrt{- \frac{1}{4 a c - b^{2}}} - b^{2} \sqrt{- \frac{1}{4 a c - b^{2}}} - b}{2 c} \right )}}{2} \]

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*sqr
t(-1/(4*a*c - b**2)) - b**2*sqrt(-1/(4*a*c - b**2)) - b)/(2*c))/2

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GIAC/XCAS [A]  time = 0.29417, size = 50, normalized size = 1.43 \[ \frac{\arctan \left (\frac{2 \, c x^{2} - b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} \]

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)