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

Optimal. Leaf size=81 \[ \frac{\log \left (2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}} \]

[Out]

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

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Rubi [A]  time = 0.113775, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{\log \left (2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.6325, size = 73, normalized size = 0.9 \[ - \frac{\log{\left (\frac{e}{2 f} + x^{2} - \frac{x \sqrt{- d}}{\sqrt{f}} \right )}}{4 \sqrt{f} \sqrt{- d}} + \frac{\log{\left (\frac{e}{2 f} + x^{2} + \frac{x \sqrt{- d}}{\sqrt{f}} \right )}}{4 \sqrt{f} \sqrt{- d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.187654, size = 191, normalized size = 2.36 \[ \frac{-\frac{\left (\sqrt{d} \sqrt{d+2 e}-d-2 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}\right )}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}-\frac{\left (\sqrt{d} \sqrt{d+2 e}+d+2 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}\right )}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{d+2 e}} \]

Antiderivative was successfully verified.

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

[Out]

(-(((-d - 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e - S
qrt[d]*Sqrt[d + 2*e]]])/Sqrt[d + e - Sqrt[d]*Sqrt[d + 2*e]]) - ((d + 2*e + Sqrt[
d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]]
])/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[d + 2*e]*Sqrt[f]
)

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Maple [B]  time = 0.07, size = 394, normalized size = 4.9 \[ -{\frac{f\sqrt{2}d}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}{\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}{\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}}+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{-df-ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}}-{\frac{f\sqrt{2}d}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}{\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}{\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{df+ef+\sqrt{d{f}^{2} \left ( d+2\,e \right ) }}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/(-d*f-e*f+(d*f^2*(d+2*e))^(1/2))^(1/2)*arct
anh(f*x*2^(1/2)/(-d*f-e*f+(d*f^2*(d+2*e))^(1/2))^(1/2))*d-1/2*f/(d*f^2*(d+2*e))^
(1/2)*2^(1/2)/(-d*f-e*f+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(-d*f-e
*f+(d*f^2*(d+2*e))^(1/2))^(1/2))*e+1/4*2^(1/2)/(-d*f-e*f+(d*f^2*(d+2*e))^(1/2))^
(1/2)*arctanh(f*x*2^(1/2)/(-d*f-e*f+(d*f^2*(d+2*e))^(1/2))^(1/2))-1/4*f/(d*f^2*(
d+2*e))^(1/2)*2^(1/2)/(d*f+e*f+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*2^(1/2)/(
d*f+e*f+(d*f^2*(d+2*e))^(1/2))^(1/2))*d-1/2*f/(d*f^2*(d+2*e))^(1/2)*2^(1/2)/(d*f
+e*f+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*2^(1/2)/(d*f+e*f+(d*f^2*(d+2*e))^(1
/2))^(1/2))*e-1/4*2^(1/2)/(d*f+e*f+(d*f^2*(d+2*e))^(1/2))^(1/2)*arctan(f*x*2^(1/
2)/(d*f+e*f+(d*f^2*(d+2*e))^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, f x^{2} - e}{4 \, f^{2} x^{4} + 4 \, d f x^{2} + 4 \, e f x^{2} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((2*f*x^2 - e)/(4*f^2*x^4 + 4*d*f*x^2 + 4*e*f*x^2 + e^2), x)

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Fricas [A]  time = 0.300178, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (-\frac{8 \, d f^{2} x^{3} + 4 \, d e f x -{\left (4 \, f^{2} x^{4} - 4 \,{\left (d - e\right )} f x^{2} + e^{2}\right )} \sqrt{-d f}}{4 \, f^{2} x^{4} + 4 \,{\left (d + e\right )} f x^{2} + e^{2}}\right )}{4 \, \sqrt{-d f}}, -\frac{\arctan \left (\frac{\sqrt{d f} x}{d}\right ) - \arctan \left (\frac{2 \, f^{2} x^{3} +{\left (2 \, d + e\right )} f x}{\sqrt{d f} e}\right )}{2 \, \sqrt{d f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.45002, size = 70, normalized size = 0.86 \[ \frac{\sqrt{- \frac{1}{d f}} \log{\left (- d x \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{2} \right )}}{4} - \frac{\sqrt{- \frac{1}{d f}} \log{\left (d x \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(d*f))*log(-d*x*sqrt(-1/(d*f)) + e/(2*f) + x**2)/4 - sqrt(-1/(d*f))*log(
d*x*sqrt(-1/(d*f)) + e/(2*f) + x**2)/4

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GIAC/XCAS [A]  time = 0.761543, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done