3.1534 \(\int \frac{1}{x^2 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=37 \[ \frac{3}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{x} \]

[Out]

-(Sqrt[1 + x^8]/x) + (3*x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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Rubi [A]  time = 0.0352554, antiderivative size = 37, 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 \[ \frac{3}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[1 + x^8]),x]

[Out]

-(Sqrt[1 + x^8]/x) + (3*x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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Rubi in Sympy [A]  time = 3.99246, size = 27, normalized size = 0.73 \[ \frac{3 x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{8} \\ \frac{15}{8} \end{matrix}\middle |{- x^{8}} \right )}}{7} - \frac{\sqrt{x^{8} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**7*hyper((1/2, 7/8), (15/8,), -x**8)/7 - sqrt(x**8 + 1)/x

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Mathematica [A]  time = 0.0322194, size = 37, normalized size = 1. \[ \frac{3}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[1 + x^8]),x]

[Out]

-(Sqrt[1 + x^8]/x) + (3*x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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Maple [A]  time = 0.038, size = 30, normalized size = 0.8 \[{\frac{3\,{x}^{7}}{7}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{7}{8}};\,{\frac{15}{8}};\,-{x}^{8})}}-{\frac{1}{x}\sqrt{{x}^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^8+1)^(1/2),x)

[Out]

3/7*x^7*hypergeom([1/2,7/8],[15/8],-x^8)-(x^8+1)^(1/2)/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{8} + 1} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^2),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^8 + 1)*x^2), x)

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Sympy [A]  time = 2.02892, size = 31, normalized size = 0.84 \[ \frac{\Gamma \left (- \frac{1}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{8}, \frac{1}{2} \\ \frac{7}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x \Gamma \left (\frac{7}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-1/8)*hyper((-1/8, 1/2), (7/8,), x**8*exp_polar(I*pi))/(8*x*gamma(7/8))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^2), x)