Optimal. Leaf size=29 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]
[Out]
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Rubi [A] time = 0.03091, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[x/(a + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 4.70512, size = 26, normalized size = 0.9 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.00939374, size = 29, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + c*x^4),x]
[Out]
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Maple [A] time = 0.002, size = 19, normalized size = 0.7 \[{\frac{1}{2}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(c*x^4+a),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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241088, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right )}{4 \, \sqrt{-a c}}, -\frac{\arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right )}{2 \, \sqrt{a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.406543, size = 56, normalized size = 1.93 \[ - \frac{\sqrt{- \frac{1}{a c}} \log{\left (- a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} + \frac{\sqrt{- \frac{1}{a c}} \log{\left (a \sqrt{- \frac{1}{a c}} + x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.223509, size = 24, normalized size = 0.83 \[ \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^4 + a),x, algorithm="giac")
[Out]