Optimal. Leaf size=25 \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
[Out]
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Rubi [A] time = 0.0317372, antiderivative size = 25, 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 \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
[In] Int[x^5/Sqrt[1 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 4.45925, size = 19, normalized size = 0.76 \[ \frac{x^{2} \sqrt{x^{4} + 1}}{4} - \frac{\operatorname{asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(x**4+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0141762, size = 25, normalized size = 1. \[ \frac{1}{4} x^2 \sqrt{x^4+1}-\frac{1}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^5/Sqrt[1 + x^4],x]
[Out]
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Maple [A] time = 0.014, size = 20, normalized size = 0.8 \[ -{\frac{{\it Arcsinh} \left ({x}^{2} \right ) }{4}}+{\frac{{x}^{2}}{4}\sqrt{{x}^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(x^4+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.44208, size = 78, normalized size = 3.12 \[ \frac{\sqrt{x^{4} + 1}}{4 \, x^{2}{\left (\frac{x^{4} + 1}{x^{4}} - 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258263, size = 117, normalized size = 4.68 \[ -\frac{2 \, x^{8} + 2 \, x^{4} -{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 1} x^{2} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) -{\left (2 \, x^{6} + x^{2}\right )} \sqrt{x^{4} + 1}}{4 \,{\left (2 \, x^{4} - 2 \, \sqrt{x^{4} + 1} x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.86892, size = 19, normalized size = 0.76 \[ \frac{x^{2} \sqrt{x^{4} + 1}}{4} - \frac{\operatorname{asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(x**4+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222529, size = 39, normalized size = 1.56 \[ \frac{1}{4} \, \sqrt{x^{4} + 1} x^{2} + \frac{1}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^4 + 1),x, algorithm="giac")
[Out]