Optimal. Leaf size=45 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.216194, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x/(x + Sqrt[x^6]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{x}{2} + \frac{\sqrt{x^{6}}}{6} + \frac{\operatorname{atan}{\left (x \right )}}{2} + \int \frac{x - \sqrt{x^{6}}}{x}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x - 1}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x + 1}\, dx - \frac{\sqrt{x^{6}}}{2 x^{2}} + \frac{\sqrt{x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(x+(x**6)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0681589, size = 0, normalized size = 0. \[ \int \frac{x}{x+\sqrt{x^6}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/(x + Sqrt[x^6]),x]
[Out]
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Maple [A] time = 0.012, size = 27, normalized size = 0.6 \[{1\arctan \left ( \sqrt{{\frac{1}{{x}^{3}}\sqrt{{x}^{6}}}}x \right ){\frac{1}{\sqrt{{\frac{1}{{x}^{3}}\sqrt{{x}^{6}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(x+(x^6)^(1/2)),x)
[Out]
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Maxima [A] time = 0.833021, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x + sqrt(x^6)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26283, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x + sqrt(x^6)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.180629, size = 2, normalized size = 0.04 \[ \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x+(x**6)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.264695, size = 16, normalized size = 0.36 \[ \frac{\arctan \left (x \sqrt{{\rm sign}\left (x\right )}\right )}{\sqrt{{\rm sign}\left (x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x + sqrt(x^6)),x, algorithm="giac")
[Out]