3.923 \(\int \frac{1}{x^7 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{x^4+1}}{3 x^2}-\frac{\sqrt{x^4+1}}{6 x^6} \]

[Out]

-Sqrt[1 + x^4]/(6*x^6) + Sqrt[1 + x^4]/(3*x^2)

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Rubi [A]  time = 0.0256047, 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 \[ \frac{\sqrt{x^4+1}}{3 x^2}-\frac{\sqrt{x^4+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(6*x^6) + Sqrt[1 + x^4]/(3*x^2)

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Rubi in Sympy [A]  time = 3.13653, size = 26, normalized size = 0.79 \[ \frac{\sqrt{x^{4} + 1}}{3 x^{2}} - \frac{\sqrt{x^{4} + 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**4+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0116528, size = 25, normalized size = 0.76 \[ \left (\frac{1}{3 x^2}-\frac{1}{6 x^6}\right ) \sqrt{x^4+1} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

(-1/(6*x^6) + 1/(3*x^2))*Sqrt[1 + x^4]

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Maple [A]  time = 0.005, size = 20, normalized size = 0.6 \[{\frac{2\,{x}^{4}-1}{6\,{x}^{6}}\sqrt{{x}^{4}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^4+1)^(1/2),x)

[Out]

1/6*(x^4+1)^(1/2)*(2*x^4-1)/x^6

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Maxima [A]  time = 1.44203, size = 34, normalized size = 1.03 \[ \frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250516, size = 70, normalized size = 2.12 \[ \frac{3 \, x^{4} - 3 \, \sqrt{x^{4} + 1} x^{2} + 1}{6 \,{\left (4 \, x^{12} + 3 \, x^{8} -{\left (4 \, x^{10} + x^{6}\right )} \sqrt{x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^7),x, algorithm="fricas")

[Out]

1/6*(3*x^4 - 3*sqrt(x^4 + 1)*x^2 + 1)/(4*x^12 + 3*x^8 - (4*x^10 + x^6)*sqrt(x^4
+ 1))

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Sympy [A]  time = 2.92878, size = 26, normalized size = 0.79 \[ \frac{\sqrt{1 + \frac{1}{x^{4}}}}{3} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**4+1)**(1/2),x)

[Out]

sqrt(1 + x**(-4))/3 - sqrt(1 + x**(-4))/(6*x**4)

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GIAC/XCAS [A]  time = 0.221168, size = 26, normalized size = 0.79 \[ -\frac{1}{6} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^4 + 1)*x^7),x, algorithm="giac")

[Out]

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