3.924 \(\int \frac{1}{x^{11} \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{x^4+1}}{10 x^{10}}+\frac{2 \sqrt{x^4+1}}{15 x^6}-\frac{4 \sqrt{x^4+1}}{15 x^2} \]

[Out]

-Sqrt[1 + x^4]/(10*x^10) + (2*Sqrt[1 + x^4])/(15*x^6) - (4*Sqrt[1 + x^4])/(15*x^
2)

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Rubi [A]  time = 0.0386037, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^4+1}}{10 x^{10}}+\frac{2 \sqrt{x^4+1}}{15 x^6}-\frac{4 \sqrt{x^4+1}}{15 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(10*x^10) + (2*Sqrt[1 + x^4])/(15*x^6) - (4*Sqrt[1 + x^4])/(15*x^
2)

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Rubi in Sympy [A]  time = 4.16691, size = 42, normalized size = 0.86 \[ - \frac{4 \sqrt{x^{4} + 1}}{15 x^{2}} + \frac{2 \sqrt{x^{4} + 1}}{15 x^{6}} - \frac{\sqrt{x^{4} + 1}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(x**4+1)**(1/2),x)

[Out]

-4*sqrt(x**4 + 1)/(15*x**2) + 2*sqrt(x**4 + 1)/(15*x**6) - sqrt(x**4 + 1)/(10*x*
*10)

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Mathematica [A]  time = 0.0147026, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^4+1} \left (8 x^8-4 x^4+3\right )}{30 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*Sqrt[1 + x^4]),x]

[Out]

-(Sqrt[1 + x^4]*(3 - 4*x^4 + 8*x^8))/(30*x^10)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{8\,{x}^{8}-4\,{x}^{4}+3}{30\,{x}^{10}}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(x^4+1)^(1/2),x)

[Out]

-1/30*(x^4+1)^(1/2)*(8*x^8-4*x^4+3)/x^10

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Maxima [A]  time = 1.43605, size = 50, normalized size = 1.02 \[ -\frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} + \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{3 \, x^{6}} - \frac{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}{10 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^11),x, algorithm="maxima")

[Out]

-1/2*sqrt(x^4 + 1)/x^2 + 1/3*(x^4 + 1)^(3/2)/x^6 - 1/10*(x^4 + 1)^(5/2)/x^10

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Fricas [A]  time = 0.255531, size = 101, normalized size = 2.06 \[ \frac{40 \, x^{8} + 35 \, x^{4} - 5 \,{\left (8 \, x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 3}{30 \,{\left (16 \, x^{20} + 20 \, x^{16} + 5 \, x^{12} -{\left (16 \, x^{18} + 12 \, x^{14} + x^{10}\right )} \sqrt{x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^11),x, algorithm="fricas")

[Out]

1/30*(40*x^8 + 35*x^4 - 5*(8*x^6 + 3*x^2)*sqrt(x^4 + 1) + 3)/(16*x^20 + 20*x^16
+ 5*x^12 - (16*x^18 + 12*x^14 + x^10)*sqrt(x^4 + 1))

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Sympy [A]  time = 6.35461, size = 44, normalized size = 0.9 \[ - \frac{4 \sqrt{1 + \frac{1}{x^{4}}}}{15} + \frac{2 \sqrt{1 + \frac{1}{x^{4}}}}{15 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{10 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(x**4+1)**(1/2),x)

[Out]

-4*sqrt(1 + x**(-4))/15 + 2*sqrt(1 + x**(-4))/(15*x**4) - sqrt(1 + x**(-4))/(10*
x**8)

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GIAC/XCAS [A]  time = 0.231434, size = 38, normalized size = 0.78 \[ -\frac{1}{10} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^11),x, algorithm="giac")

[Out]

-1/10*(1/x^4 + 1)^(5/2) + 1/3*(1/x^4 + 1)^(3/2) - 1/2*sqrt(1/x^4 + 1)