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

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

[Out]

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

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Rubi [A]  time = 0.103538, 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 \[ \frac{\log \left (x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}}-\frac{\log \left (-x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.4696, size = 66, normalized size = 0.85 \[ - \frac{\log{\left (\frac{d}{e} + x^{2} - \frac{x \sqrt{- b + 2 d e}}{e} \right )}}{2 \sqrt{- b + 2 d e}} + \frac{\log{\left (\frac{d}{e} + x^{2} + \frac{x \sqrt{- b + 2 d e}}{e} \right )}}{2 \sqrt{- b + 2 d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.22723, size = 182, normalized size = 2.33 \[ \frac{\frac{\left (-\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}\right )}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}-\frac{\left (\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}\right )}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 88, normalized size = 1.1 \[ -{\frac{1}{-4\,de+2\,b}\sqrt{2\,de-b}\ln \left ( d+e{x}^{2}+x\sqrt{2\,de-b} \right ) }+{\frac{1}{-4\,de+2\,b}\sqrt{2\,de-b}\ln \left ( -e{x}^{2}+x\sqrt{2\,de-b}-d \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(-4*d*e+2*b)*(2*d*e-b)^(1/2)*ln(d+e*x^2+x*(2*d*e-b)^(1/2))+1/(-4*d*e+2*b)*(2*
d*e-b)^(1/2)*ln(-e*x^2+x*(2*d*e-b)^(1/2)-d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289879, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{2 \,{\left (2 \, d e^{2} - b e\right )} x^{3} + 2 \,{\left (2 \, d^{2} e - b d\right )} x +{\left (e^{2} x^{4} +{\left (4 \, d e - b\right )} x^{2} + d^{2}\right )} \sqrt{2 \, d e - b}}{e^{2} x^{4} + b x^{2} + d^{2}}\right )}{2 \, \sqrt{2 \, d e - b}}, \frac{\arctan \left (\frac{\sqrt{-2 \, d e + b} e x}{2 \, d e - b}\right ) + \arctan \left (\frac{e^{2} x^{3} -{\left (d e - b\right )} x}{\sqrt{-2 \, d e + b} d}\right )}{\sqrt{-2 \, d e + b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.15032, size = 121, normalized size = 1.55 \[ \frac{\sqrt{- \frac{1}{b - 2 d e}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- b \sqrt{- \frac{1}{b - 2 d e}} + 2 d e \sqrt{- \frac{1}{b - 2 d e}}\right )}{e} \right )}}{2} - \frac{\sqrt{- \frac{1}{b - 2 d e}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (b \sqrt{- \frac{1}{b - 2 d e}} - 2 d e \sqrt{- \frac{1}{b - 2 d e}}\right )}{e} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.453287, 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 + b*x^2 + d^2),x, algorithm="giac")

[Out]

Done