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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]