3.1520 \(\int \frac{1}{x^{13} \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{x^8+1}}{6 x^4}-\frac{\sqrt{x^8+1}}{12 x^{12}} \]

[Out]

-Sqrt[1 + x^8]/(12*x^12) + Sqrt[1 + x^8]/(6*x^4)

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Rubi [A]  time = 0.026796, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{x^8+1}}{6 x^4}-\frac{\sqrt{x^8+1}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^13*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(12*x^12) + Sqrt[1 + x^8]/(6*x^4)

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Rubi in Sympy [A]  time = 3.26514, size = 26, normalized size = 0.79 \[ \frac{\sqrt{x^{8} + 1}}{6 x^{4}} - \frac{\sqrt{x^{8} + 1}}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**13/(x**8+1)**(1/2),x)

[Out]

sqrt(x**8 + 1)/(6*x**4) - sqrt(x**8 + 1)/(12*x**12)

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Mathematica [A]  time = 0.0130553, size = 25, normalized size = 0.76 \[ \left (\frac{1}{6 x^4}-\frac{1}{12 x^{12}}\right ) \sqrt{x^8+1} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^13*Sqrt[1 + x^8]),x]

[Out]

(-1/(12*x^12) + 1/(6*x^4))*Sqrt[1 + x^8]

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Maple [A]  time = 0.005, size = 20, normalized size = 0.6 \[{\frac{2\,{x}^{8}-1}{12\,{x}^{12}}\sqrt{{x}^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^13/(x^8+1)^(1/2),x)

[Out]

1/12*(x^8+1)^(1/2)*(2*x^8-1)/x^12

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Maxima [A]  time = 1.42842, size = 34, normalized size = 1.03 \[ \frac{\sqrt{x^{8} + 1}}{4 \, x^{4}} - \frac{{\left (x^{8} + 1\right )}^{\frac{3}{2}}}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^13),x, algorithm="maxima")

[Out]

1/4*sqrt(x^8 + 1)/x^4 - 1/12*(x^8 + 1)^(3/2)/x^12

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Fricas [A]  time = 0.22847, size = 70, normalized size = 2.12 \[ \frac{3 \, x^{8} - 3 \, \sqrt{x^{8} + 1} x^{4} + 1}{12 \,{\left (4 \, x^{24} + 3 \, x^{16} -{\left (4 \, x^{20} + x^{12}\right )} \sqrt{x^{8} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^13),x, algorithm="fricas")

[Out]

1/12*(3*x^8 - 3*sqrt(x^8 + 1)*x^4 + 1)/(4*x^24 + 3*x^16 - (4*x^20 + x^12)*sqrt(x
^8 + 1))

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Sympy [A]  time = 9.4109, size = 26, normalized size = 0.79 \[ \frac{\sqrt{1 + \frac{1}{x^{8}}}}{6} - \frac{\sqrt{1 + \frac{1}{x^{8}}}}{12 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**13/(x**8+1)**(1/2),x)

[Out]

sqrt(1 + x**(-8))/6 - sqrt(1 + x**(-8))/(12*x**8)

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GIAC/XCAS [A]  time = 0.233863, size = 26, normalized size = 0.79 \[ -\frac{1}{12} \,{\left (\frac{1}{x^{8}} + 1\right )}^{\frac{3}{2}} + \frac{1}{4} \, \sqrt{\frac{1}{x^{8}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^13),x, algorithm="giac")

[Out]

-1/12*(1/x^8 + 1)^(3/2) + 1/4*sqrt(1/x^8 + 1)