3.1527 \(\int \frac{1}{x^7 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\sqrt{x^8+1}}{6 x^6}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{12 \sqrt{x^8+1}} \]

[Out]

-Sqrt[1 + x^8]/(6*x^6) - ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcT
an[x^2], 1/2])/(12*Sqrt[1 + x^8])

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Rubi [A]  time = 0.0685701, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{x^8+1}}{6 x^6}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{12 \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(6*x^6) - ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcT
an[x^2], 1/2])/(12*Sqrt[1 + x^8])

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Rubi in Sympy [A]  time = 4.71733, size = 54, normalized size = 0.87 \[ - \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{8} + 1}} - \frac{\sqrt{x^{8} + 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**8+1)**(1/2),x)

[Out]

-sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*elliptic_f(2*atan(x**2), 1/2)/(12*sqr
t(x**8 + 1)) - sqrt(x**8 + 1)/(6*x**6)

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Mathematica [C]  time = 0.0258229, size = 36, normalized size = 0.58 \[ -\frac{x^8 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^8\right )+\sqrt{x^8+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[1 + x^8]),x]

[Out]

-(Sqrt[1 + x^8] + x^8*Hypergeometric2F1[1/4, 1/2, 5/4, -x^8])/(6*x^6)

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Maple [C]  time = 0.027, size = 30, normalized size = 0.5 \[ -{\frac{1}{6\,{x}^{6}}\sqrt{{x}^{8}+1}}-{\frac{{x}^{2}}{6}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^8+1)^(1/2),x)

[Out]

-1/6*(x^8+1)^(1/2)/x^6-1/6*x^2*hypergeom([1/4,1/2],[5/4],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^7),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{8} + 1} x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^7),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^8 + 1)*x^7), x)

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Sympy [A]  time = 3.1458, size = 32, normalized size = 0.52 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{6} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**8+1)**(1/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**8*exp_polar(I*pi))/(8*x**6*gamma(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{8} + 1} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^8 + 1)*x^7),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^7), x)