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

Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.128848, antiderivative size = 42, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

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]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.4651, size = 37, normalized size = 0.88 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{f} \left (e + x^{2} \left (2 d + 2 f\right )\right )}{\sqrt{d} e} \right )}}{4 \sqrt{d} e \sqrt{f}} \]

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]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0320959, size = 42, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

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]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 42, normalized size = 1. \[{\frac{1}{4\,e}\arctan \left ({\frac{2\, \left ( 4\,df+4\,{f}^{2} \right ){x}^{2}+4\,ef}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.300433, 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))/(sq
rt(d*f)*e)]

_______________________________________________________________________________________

Sympy [A]  time = 1.87182, size = 78, normalized size = 1.86 \[ \frac{- \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{2} + \frac{- d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{8} + \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{2} + \frac{d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{8}}{e} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.342735, size = 51, normalized size = 1.21 \[ \frac{\arctan \left (\frac{{\left (2 \, d f x^{2} + 2 \, f^{2} x^{2} + f e\right )} e^{\left (-1\right )}}{\sqrt{d f}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{d f}} \]

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)*e^(-1)/sqrt(d*f))*e^(-1)/sqrt(d*f)