3.1536 \(\int \frac{1}{x^2 \sqrt{1+x^8}} \, dx\)

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 = 0.0086354, 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, Rules used = {325, 364} \[ \frac{3}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{x} \]

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{1+x^8}} \, dx &=-\frac{\sqrt{1+x^8}}{x}+3 \int \frac{x^6}{\sqrt{1+x^8}} \, dx\\ &=-\frac{\sqrt{1+x^8}}{x}+\frac{3}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )\\ \end{align*}

Mathematica [A]  time = 0.0026044, size = 20, normalized size = 0.54 \[ -\frac{\, _2F_1\left (-\frac{1}{8},\frac{1}{2};\frac{7}{8};-x^8\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[1 + x^8]),x]

[Out]

-(Hypergeometric2F1[-1/8, 1/2, 7/8, -x^8]/x)

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Maple [A]  time = 0.025, size = 30, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{x^{8} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+1)^(1/2),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. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{8} + 1}}{x^{10} + x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^8 + 1)/(x^10 + x^2), x)

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Sympy [C]  time = 0.581055, size = 31, normalized size = 0.84 \begin{align*} \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 )} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^8 + 1)*x^2), x)