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

Optimal. Leaf size=73 \[ \frac{\log \left (2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}} \]

[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 = 0.0875291, antiderivative size = 73, 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 = 68.9749, size = 66, normalized size = 0.9 \[ - \frac{\log{\left (- \frac{\sqrt{d} x}{\sqrt{f}} + \frac{e}{2 f} + x^{2} \right )}}{4 \sqrt{d} \sqrt{f}} + \frac{\log{\left (\frac{\sqrt{d} x}{\sqrt{f}} + \frac{e}{2 f} + x^{2} \right )}}{4 \sqrt{d} \sqrt{f}} \]

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

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Mathematica [C]  time = 0.218115, size = 233, normalized size = 3.19 \[ \frac{-\frac{\left (\sqrt{d} \sqrt{2 e-d}-i d+2 i e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}\right )}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}-\frac{\left (\sqrt{d} \sqrt{2 e-d}+i d-2 i e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}\right )}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{2 e-d}} \]

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]

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

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Maple [B]  time = 0.069, size = 394, normalized size = 5.4 \[{\frac{f\sqrt{2}d}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{-df+ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}}+{\frac{f\sqrt{2}d}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}}+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{df-ef+\sqrt{d{f}^{2} \left ( d-2\,e \right ) }}}}} \]

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)*arcta
n(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))+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)*arctanh(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))

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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 = 0.285423, 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 = 2.51022, size = 63, 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 = 0.748517, 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