3.1522 \(\int \frac{1}{x^{13} \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{x^8+1}}{6 x^4}-\frac{\sqrt{x^8+1}}{12 x^{12}} \]

[Out]

-Sqrt[1 + x^8]/(12*x^12) + Sqrt[1 + x^8]/(6*x^4)

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Rubi [A]  time = 0.00632, 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, Rules used = {271, 264} \[ \frac{\sqrt{x^8+1}}{6 x^4}-\frac{\sqrt{x^8+1}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^13*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(12*x^12) + Sqrt[1 + x^8]/(6*x^4)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^{13} \sqrt{1+x^8}} \, dx &=-\frac{\sqrt{1+x^8}}{12 x^{12}}-\frac{2}{3} \int \frac{1}{x^5 \sqrt{1+x^8}} \, dx\\ &=-\frac{\sqrt{1+x^8}}{12 x^{12}}+\frac{\sqrt{1+x^8}}{6 x^4}\\ \end{align*}

Mathematica [A]  time = 0.004014, size = 23, normalized size = 0.7 \[ -\frac{\left (1-2 x^8\right ) \sqrt{x^8+1}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*Sqrt[1 + x^8]),x]

[Out]

-((1 - 2*x^8)*Sqrt[1 + x^8])/(12*x^12)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{8}-1}{12\,{x}^{12}}\sqrt{{x}^{8}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^13/(x^8+1)^(1/2),x)

[Out]

1/12*(x^8+1)^(1/2)*(2*x^8-1)/x^12

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Maxima [A]  time = 0.94874, size = 34, normalized size = 1.03 \begin{align*} \frac{\sqrt{x^{8} + 1}}{4 \, x^{4}} - \frac{{\left (x^{8} + 1\right )}^{\frac{3}{2}}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.32823, size = 65, normalized size = 1.97 \begin{align*} \frac{2 \, x^{12} +{\left (2 \, x^{8} - 1\right )} \sqrt{x^{8} + 1}}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/12*(2*x^12 + (2*x^8 - 1)*sqrt(x^8 + 1))/x^12

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Sympy [A]  time = 2.17836, size = 26, normalized size = 0.79 \begin{align*} \frac{\sqrt{1 + \frac{1}{x^{8}}}}{6} - \frac{\sqrt{1 + \frac{1}{x^{8}}}}{12 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**13/(x**8+1)**(1/2),x)

[Out]

sqrt(1 + x**(-8))/6 - sqrt(1 + x**(-8))/(12*x**8)

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Giac [A]  time = 1.12653, size = 26, normalized size = 0.79 \begin{align*} -\frac{1}{12} \,{\left (\frac{1}{x^{8}} + 1\right )}^{\frac{3}{2}} + \frac{1}{4} \, \sqrt{\frac{1}{x^{8}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

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