3.1514 \(\int \frac{x^{11}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]

[Out]

(x^4*Sqrt[1 + x^8])/8 - ArcSinh[x^4]/8

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Rubi [A]  time = 0.0364963, 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 \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^11/Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8])/8 - ArcSinh[x^4]/8

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Rubi in Sympy [A]  time = 4.66603, 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**11/(x**8+1)**(1/2),x)

[Out]

x**4*sqrt(x**8 + 1)/8 - asinh(x**4)/8

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Mathematica [A]  time = 0.0127126, size = 25, normalized size = 1. \[ \frac{1}{8} x^4 \sqrt{x^8+1}-\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8])/8 - ArcSinh[x^4]/8

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Maple [A]  time = 0.035, 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^11/(x^8+1)^(1/2),x)

[Out]

-1/8*arcsinh(x^4)+1/8*x^4*(x^8+1)^(1/2)

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Maxima [A]  time = 1.44347, 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(x^11/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

1/8*sqrt(x^8 + 1)/(x^4*((x^8 + 1)/x^8 - 1)) - 1/16*log(sqrt(x^8 + 1)/x^4 + 1) +
1/16*log(sqrt(x^8 + 1)/x^4 - 1)

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Fricas [A]  time = 0.221649, size = 117, normalized size = 4.68 \[ -\frac{2 \, x^{16} + 2 \, x^{8} -{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )} \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) -{\left (2 \, x^{12} + x^{4}\right )} \sqrt{x^{8} + 1}}{8 \,{\left (2 \, x^{8} - 2 \, \sqrt{x^{8} + 1} x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

-1/8*(2*x^16 + 2*x^8 - (2*x^8 - 2*sqrt(x^8 + 1)*x^4 + 1)*log(-x^4 + sqrt(x^8 + 1
)) - (2*x^12 + x^4)*sqrt(x^8 + 1))/(2*x^8 - 2*sqrt(x^8 + 1)*x^4 + 1)

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Sympy [A]  time = 8.38976, 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**11/(x**8+1)**(1/2),x)

[Out]

x**4*sqrt(x**8 + 1)/8 - asinh(x**4)/8

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GIAC/XCAS [A]  time = 0.243473, size = 50, normalized size = 2. \[ \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{16} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} + 1\right ) + \frac{1}{16} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

1/8*sqrt(x^8 + 1)*x^4 - 1/16*ln(sqrt(1/x^8 + 1) + 1) + 1/16*ln(sqrt(1/x^8 + 1) -
1)