Optimal. Leaf size=75 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]
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Rubi [A] time = 0.0971491, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)/(d^2 + e^2*x^4),x]
[Out]
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Rubi in Sympy [A] time = 12.5742, size = 73, normalized size = 0.97 \[ - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{d} \sqrt{e}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}} \right )}}{2 \sqrt{d} \sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)/(e**2*x**4+d**2),x)
[Out]
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Mathematica [A] time = 0.0510194, size = 60, normalized size = 0.8 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )-\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)/(d^2 + e^2*x^4),x]
[Out]
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Maple [B] time = 0.013, size = 290, normalized size = 3.9 \[{\frac{\sqrt{2}}{8\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ) }+{\frac{\sqrt{2}}{8\,e}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)/(e^2*x^4+d^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(e^2*x^4 + d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289028, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2} \log \left (\frac{4 \, d e^{2} x^{3} - 4 \, d^{2} e x + \sqrt{2}{\left (e^{2} x^{4} - 4 \, d e x^{2} + d^{2}\right )} \sqrt{-d e}}{e^{2} x^{4} + d^{2}}\right )}{4 \, \sqrt{-d e}}, \frac{\sqrt{2}{\left (\arctan \left (\frac{\sqrt{2} \sqrt{d e} x}{2 \, d}\right ) + \arctan \left (\frac{\sqrt{2}{\left (e^{2} x^{3} + d e x\right )}}{2 \, \sqrt{d e} d}\right )\right )}}{2 \, \sqrt{d e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(e^2*x^4 + d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.464425, size = 87, normalized size = 1.16 \[ - \frac{\sqrt{2} \sqrt{- \frac{1}{d e}} \log{\left (- \sqrt{2} d x \sqrt{- \frac{1}{d e}} - \frac{d}{e} + x^{2} \right )}}{4} + \frac{\sqrt{2} \sqrt{- \frac{1}{d e}} \log{\left (\sqrt{2} d x \sqrt{- \frac{1}{d e}} - \frac{d}{e} + x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)/(e**2*x**4+d**2),x)
[Out]
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GIAC/XCAS [A] time = 0.27754, size = 300, normalized size = 4. \[ \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} - 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} - \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (-\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)/(e^2*x^4 + d^2),x, algorithm="giac")
[Out]