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

Optimal. Leaf size=78 \[ \frac{\log \left (x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}}-\frac{\log \left (-x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}} \]

[Out]

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

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Rubi [A]  time = 0.0891105, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\log \left (x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}}-\frac{\log \left (-x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.5672, size = 66, normalized size = 0.85 \[ - \frac{\log{\left (\frac{d}{e} + x^{2} - \frac{x \sqrt{2 d e - f}}{e} \right )}}{2 \sqrt{2 d e - f}} + \frac{\log{\left (\frac{d}{e} + x^{2} + \frac{x \sqrt{2 d e - f}}{e} \right )}}{2 \sqrt{2 d e - f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.239329, size = 182, normalized size = 2.33 \[ \frac{\frac{\left (-\sqrt{f^2-4 d^2 e^2}+2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right )}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}-\frac{\left (\sqrt{f^2-4 d^2 e^2}+2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 69, normalized size = 0.9 \[ -{\frac{1}{2}\ln \left ( -e{x}^{2}+x\sqrt{2\,de-f}-d \right ){\frac{1}{\sqrt{2\,de-f}}}}+{\frac{1}{2}\ln \left ( d+e{x}^{2}+x\sqrt{2\,de-f} \right ){\frac{1}{\sqrt{2\,de-f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(2*d*e-f)^(1/2)*ln(-e*x^2+x*(2*d*e-f)^(1/2)-d)+1/2*ln(d+e*x^2+x*(2*d*e-f)^(
1/2))/(2*d*e-f)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289568, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{2 \,{\left (2 \, d e^{2} - e f\right )} x^{3} + 2 \,{\left (2 \, d^{2} e - d f\right )} x +{\left (e^{2} x^{4} +{\left (4 \, d e - f\right )} x^{2} + d^{2}\right )} \sqrt{2 \, d e - f}}{e^{2} x^{4} + f x^{2} + d^{2}}\right )}{2 \, \sqrt{2 \, d e - f}}, -\frac{\arctan \left (-\frac{\sqrt{-2 \, d e + f} e x}{2 \, d e - f}\right ) - \arctan \left (\frac{e^{2} x^{3} -{\left (d e - f\right )} x}{\sqrt{-2 \, d e + f} d}\right )}{\sqrt{-2 \, d e + f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.27549, size = 110, normalized size = 1.41 \[ - \frac{\sqrt{\frac{1}{2 d e - f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- 2 d e \sqrt{\frac{1}{2 d e - f}} + f \sqrt{\frac{1}{2 d e - f}}\right )}{e} \right )}}{2} + \frac{\sqrt{\frac{1}{2 d e - f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (2 d e \sqrt{\frac{1}{2 d e - f}} - f \sqrt{\frac{1}{2 d e - f}}\right )}{e} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.454, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done