Optimal. Leaf size=34 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
[Out]
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Rubi [A] time = 0.0372819, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In] Int[(b*x^2 + c*x^4)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 11.274, size = 29, normalized size = 0.85 \[ - \frac{1}{b x} - \frac{\sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0217809, size = 34, normalized size = 1. \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^2 + c*x^4)^(-1),x]
[Out]
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Maple [A] time = 0.005, size = 30, normalized size = 0.9 \[ -{\frac{c}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{1}{bx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^4+b*x^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(1/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25905, size = 1, normalized size = 0.03 \[ \left [\frac{x \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right ) + 1}{b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.27202, size = 65, normalized size = 1.91 \[ \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (- \frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{b^{3}}} \log{\left (\frac{b^{2} \sqrt{- \frac{c}{b^{3}}}}{c} + x \right )}}{2} - \frac{1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.269403, size = 39, normalized size = 1.15 \[ -\frac{c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b} - \frac{1}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]