3.1521 \(\int \frac{1}{x^{17} \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{16 x^{16}} \]

[Out]

-Sqrt[1 + x^8]/(16*x^16) + (3*Sqrt[1 + x^8])/(32*x^8) - (3*ArcTanh[Sqrt[1 + x^8]
])/32

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Rubi [A]  time = 0.0528254, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right )-\frac{\sqrt{x^8+1}}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^17*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(16*x^16) + (3*Sqrt[1 + x^8])/(32*x^8) - (3*ArcTanh[Sqrt[1 + x^8]
])/32

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Rubi in Sympy [A]  time = 5.06273, size = 41, normalized size = 0.87 \[ - \frac{3 \operatorname{atanh}{\left (\sqrt{x^{8} + 1} \right )}}{32} + \frac{3 \sqrt{x^{8} + 1}}{32 x^{8}} - \frac{\sqrt{x^{8} + 1}}{16 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**17/(x**8+1)**(1/2),x)

[Out]

-3*atanh(sqrt(x**8 + 1))/32 + 3*sqrt(x**8 + 1)/(32*x**8) - sqrt(x**8 + 1)/(16*x*
*16)

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Mathematica [A]  time = 0.0433811, size = 37, normalized size = 0.79 \[ \frac{1}{32} \left (\frac{\sqrt{x^8+1} \left (3 x^8-2\right )}{x^{16}}-3 \tanh ^{-1}\left (\sqrt{x^8+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^17*Sqrt[1 + x^8]),x]

[Out]

((Sqrt[1 + x^8]*(-2 + 3*x^8))/x^16 - 3*ArcTanh[Sqrt[1 + x^8]])/32

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Maple [A]  time = 0.037, size = 42, normalized size = 0.9 \[{\frac{3\,{x}^{16}+{x}^{8}-2}{32\,{x}^{16}}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{3}{32}\ln \left ({1 \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^17/(x^8+1)^(1/2),x)

[Out]

1/32*(3*x^16+x^8-2)/x^16/(x^8+1)^(1/2)+3/32*ln(((x^8+1)^(1/2)-1)/(x^8)^(1/2))

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Maxima [A]  time = 1.43816, size = 86, normalized size = 1.83 \[ -\frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \,{\left (2 \, x^{8} -{\left (x^{8} + 1\right )}^{2} + 1\right )}} - \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="maxima")

[Out]

-1/32*(3*(x^8 + 1)^(3/2) - 5*sqrt(x^8 + 1))/(2*x^8 - (x^8 + 1)^2 + 1) - 3/64*log
(sqrt(x^8 + 1) + 1) + 3/64*log(sqrt(x^8 + 1) - 1)

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Fricas [A]  time = 0.237902, size = 70, normalized size = 1.49 \[ -\frac{3 \, x^{16} \log \left (\sqrt{x^{8} + 1} + 1\right ) - 3 \, x^{16} \log \left (\sqrt{x^{8} + 1} - 1\right ) - 2 \,{\left (3 \, x^{8} - 2\right )} \sqrt{x^{8} + 1}}{64 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="fricas")

[Out]

-1/64*(3*x^16*log(sqrt(x^8 + 1) + 1) - 3*x^16*log(sqrt(x^8 + 1) - 1) - 2*(3*x^8
- 2)*sqrt(x^8 + 1))/x^16

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Sympy [A]  time = 27.263, size = 60, normalized size = 1.28 \[ - \frac{3 \operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{32} + \frac{3}{32 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} + \frac{1}{32 x^{12} \sqrt{1 + \frac{1}{x^{8}}}} - \frac{1}{16 x^{20} \sqrt{1 + \frac{1}{x^{8}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**17/(x**8+1)**(1/2),x)

[Out]

-3*asinh(x**(-4))/32 + 3/(32*x**4*sqrt(1 + x**(-8))) + 1/(32*x**12*sqrt(1 + x**(
-8))) - 1/(16*x**20*sqrt(1 + x**(-8)))

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GIAC/XCAS [A]  time = 0.228642, size = 66, normalized size = 1.4 \[ \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \, x^{16}} - \frac{3}{64} \,{\rm ln}\left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \,{\rm ln}\left (\sqrt{x^{8} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^17),x, algorithm="giac")

[Out]

1/32*(3*(x^8 + 1)^(3/2) - 5*sqrt(x^8 + 1))/x^16 - 3/64*ln(sqrt(x^8 + 1) + 1) + 3
/64*ln(sqrt(x^8 + 1) - 1)