Optimal. Leaf size=70 \[ \frac{\log \left (x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}}-\frac{\log \left (-x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}} \]
[Out]
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Rubi [A] time = 0.0922556, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\log \left (x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}}-\frac{\log \left (-x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}} \]
Antiderivative was successfully verified.
[In] Int[(d - e*x^2)/(d^2 - f*x^2 + e^2*x^4),x]
[Out]
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Rubi in Sympy [A] time = 32.2335, size = 66, normalized size = 0.94 \[ - \frac{\log{\left (\frac{d}{e} + x^{2} - \frac{x \sqrt{2 d e + f}}{e} \right )}}{2 \sqrt{2 d e + f}} + \frac{\log{\left (\frac{d}{e} + x^{2} + \frac{x \sqrt{2 d e + f}}{e} \right )}}{2 \sqrt{2 d e + f}} \]
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]
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Mathematica [B] time = 0.23142, size = 190, normalized size = 2.71 \[ \frac{\frac{\left (-\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}-\frac{\left (\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d - e*x^2)/(d^2 - f*x^2 + e^2*x^4),x]
[Out]
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Maple [A] time = 0.022, size = 61, normalized size = 0.9 \[{\frac{1}{2}\ln \left ( d+e{x}^{2}+x\sqrt{2\,de+f} \right ){\frac{1}{\sqrt{2\,de+f}}}}-{\frac{1}{2}\ln \left ( -e{x}^{2}+x\sqrt{2\,de+f}-d \right ){\frac{1}{\sqrt{2\,de+f}}}} \]
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]
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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]
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Fricas [A] time = 0.29136, 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]
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Sympy [A] time = 2.20474, size = 112, normalized size = 1.6 \[ - \frac{\sqrt{\frac{1}{2 d e + f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- 2 d e \sqrt{\frac{1}{2 d e + f}} - f \sqrt{\frac{1}{2 d e + f}}\right )}{e} \right )}}{2} + \frac{\sqrt{\frac{1}{2 d e + f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (2 d e \sqrt{\frac{1}{2 d e + f}} + f \sqrt{\frac{1}{2 d e + f}}\right )}{e} \right )}}{2} \]
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]
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GIAC/XCAS [A] time = 0.441338, 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]