Optimal. Leaf size=25 \[ \frac{1}{8} \sinh ^{-1}\left (x^4\right )+\frac{1}{8} \sqrt{x^8+1} x^4 \]
[Out]
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Rubi [A] time = 0.026205, 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 \[ \frac{1}{8} \sinh ^{-1}\left (x^4\right )+\frac{1}{8} \sqrt{x^8+1} x^4 \]
Antiderivative was successfully verified.
[In] Int[x^3*Sqrt[1 + x^8],x]
[Out]
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Rubi in Sympy [A] time = 2.72855, size = 19, normalized size = 0.76 \[ \frac{x^{4} \sqrt{x^{8} + 1}}{8} + \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(x**8+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0129308, size = 22, normalized size = 0.88 \[ \frac{1}{8} \left (\sinh ^{-1}\left (x^4\right )+\sqrt{x^8+1} x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3*Sqrt[1 + x^8],x]
[Out]
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Maple [A] time = 0.044, size = 20, normalized size = 0.8 \[{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{8}}+{\frac{{x}^{4}}{8}\sqrt{{x}^{8}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(x^8+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.42931, size = 78, normalized size = 3.12 \[ \frac{\sqrt{x^{8} + 1}}{8 \, x^{4}{\left (\frac{x^{8} + 1}{x^{8}} - 1\right )}} + \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221807, size = 116, normalized size = 4.64 \[ -\frac{2 \, x^{16} + 2 \, x^{8} +{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )} \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) -{\left (2 \, x^{12} + x^{4}\right )} \sqrt{x^{8} + 1}}{8 \,{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.13698, size = 19, normalized size = 0.76 \[ \frac{x^{4} \sqrt{x^{8} + 1}}{8} + \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228873, size = 39, normalized size = 1.56 \[ \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{8} \,{\rm ln}\left (-x^{4} + \sqrt{x^{8} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x^3,x, algorithm="giac")
[Out]