Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
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Rubi [A] time = 0.050205, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[x/(a + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 6.21374, size = 39, normalized size = 0.8 \[ \frac{x^{2}}{4 a \left (a + c x^{4}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(c*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.0511806, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{c}}+\frac{x^2}{4 a \left (a+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.008, size = 40, normalized size = 0.8 \[{\frac{{x}^{2}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{1}{4\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(c*x^4+a)^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(x/(c*x^4 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247717, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a c} x^{2} +{\left (c x^{4} + a\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right )}{8 \,{\left (a c x^{4} + a^{2}\right )} \sqrt{-a c}}, \frac{\sqrt{a c} x^{2} -{\left (c x^{4} + a\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right )}{4 \,{\left (a c x^{4} + a^{2}\right )} \sqrt{a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.85093, size = 83, normalized size = 1.69 \[ \frac{x^{2}}{4 a^{2} + 4 a c x^{4}} - \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} + \frac{\sqrt{- \frac{1}{a^{3} c}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} c}} + x^{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218429, size = 53, normalized size = 1.08 \[ \frac{x^{2}}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a)^2,x, algorithm="giac")
[Out]