3.370 \(\int \frac{x}{e^2+4 e f x^2-4 d f x^4+4 f^2 x^4} \, dx\)

Optimal. Leaf size=44 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (e-2 x^2 (d-f)\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

[Out]

-ArcTanh[(Sqrt[f]*(e - 2*(d - f)*x^2))/(Sqrt[d]*e)]/(4*Sqrt[d]*e*Sqrt[f])

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Rubi [A]  time = 0.134713, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (e-2 x^2 (d-f)\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Int[x/(e^2 + 4*e*f*x^2 - 4*d*f*x^4 + 4*f^2*x^4),x]

[Out]

-ArcTanh[(Sqrt[f]*(e - 2*(d - f)*x^2))/(Sqrt[d]*e)]/(4*Sqrt[d]*e*Sqrt[f])

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Rubi in Sympy [A]  time = 44.4472, size = 39, normalized size = 0.89 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{f} \left (e + x^{2} \left (- 2 d + 2 f\right )\right )}{\sqrt{d} e} \right )}}{4 \sqrt{d} e \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-4*d*f*x**4+4*f**2*x**4+4*e*f*x**2+e**2),x)

[Out]

-atanh(sqrt(f)*(e + x**2*(-2*d + 2*f))/(sqrt(d)*e))/(4*sqrt(d)*e*sqrt(f))

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Mathematica [A]  time = 0.0356423, size = 46, normalized size = 1.05 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (-2 d x^2+e+2 f x^2\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(e^2 + 4*e*f*x^2 - 4*d*f*x^4 + 4*f^2*x^4),x]

[Out]

-ArcTanh[(Sqrt[f]*(e - 2*d*x^2 + 2*f*x^2))/(Sqrt[d]*e)]/(4*Sqrt[d]*e*Sqrt[f])

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Maple [A]  time = 0.004, size = 42, normalized size = 1. \[{\frac{1}{4\,e}{\it Artanh} \left ({\frac{2\, \left ( 4\,df-4\,{f}^{2} \right ){x}^{2}-4\,ef}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-4*d*f*x^4+4*f^2*x^4+4*e*f*x^2+e^2),x)

[Out]

1/4/e/(d*f)^(1/2)*arctanh(1/4*(2*(4*d*f-4*f^2)*x^2-4*e*f)/e/(d*f)^(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/(4*d*f*x^4 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284293, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \, d e^{2} f - 4 \,{\left (d^{2} e f - d e f^{2}\right )} x^{2} -{\left (4 \,{\left (d^{2} f - 2 \, d f^{2} + f^{3}\right )} x^{4} + d e^{2} + e^{2} f - 4 \,{\left (d e f - e f^{2}\right )} x^{2}\right )} \sqrt{d f}}{4 \,{\left (d f - f^{2}\right )} x^{4} - 4 \, e f x^{2} - e^{2}}\right )}{8 \, \sqrt{d f} e}, -\frac{\arctan \left (-\frac{{\left (2 \,{\left (d - f\right )} x^{2} - e\right )} \sqrt{-d f}}{d e}\right )}{4 \, \sqrt{-d f} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(4*d*f*x^4 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="fricas")

[Out]

[1/8*log((2*d*e^2*f - 4*(d^2*e*f - d*e*f^2)*x^2 - (4*(d^2*f - 2*d*f^2 + f^3)*x^4
+ d*e^2 + e^2*f - 4*(d*e*f - e*f^2)*x^2)*sqrt(d*f))/(4*(d*f - f^2)*x^4 - 4*e*f*
x^2 - e^2))/(sqrt(d*f)*e), -1/4*arctan(-(2*(d - f)*x^2 - e)*sqrt(-d*f)/(d*e))/(s
qrt(-d*f)*e)]

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Sympy [A]  time = 2.00816, size = 75, normalized size = 1.7 \[ - \frac{\frac{\sqrt{\frac{1}{d f}} \log{\left (x^{2} + \frac{- d e \sqrt{\frac{1}{d f}} - e}{2 d - 2 f} \right )}}{8} - \frac{\sqrt{\frac{1}{d f}} \log{\left (x^{2} + \frac{d e \sqrt{\frac{1}{d f}} - e}{2 d - 2 f} \right )}}{8}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-4*d*f*x**4+4*f**2*x**4+4*e*f*x**2+e**2),x)

[Out]

-(sqrt(1/(d*f))*log(x**2 + (-d*e*sqrt(1/(d*f)) - e)/(2*d - 2*f))/8 - sqrt(1/(d*f
))*log(x**2 + (d*e*sqrt(1/(d*f)) - e)/(2*d - 2*f))/8)/e

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GIAC/XCAS [A]  time = 0.346464, size = 55, normalized size = 1.25 \[ -\frac{\arctan \left (\frac{2 \, d f x^{2} - 2 \, f^{2} x^{2} - f e}{\sqrt{-d f e^{2}}}\right )}{4 \, \sqrt{-d f e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(4*d*f*x^4 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="giac")

[Out]

-1/4*arctan((2*d*f*x^2 - 2*f^2*x^2 - f*e)/sqrt(-d*f*e^2))/sqrt(-d*f*e^2)