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]

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

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 0.70 \[ -\frac {\left (1-2 x^8\right ) \sqrt {x^8+1}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^13*Sqrt[1 + x^8]),x]

[Out]

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

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fricas [A]  time = 0.80, size = 26, normalized size = 0.79 \[ \frac {2 \, x^{12} + {\left (2 \, x^{8} - 1\right )} \sqrt {x^{8} + 1}}{12 \, x^{12}} \]

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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giac [A]  time = 0.17, size = 40, normalized size = 1.21 \[ \frac {3 \, {\left (x^{4} - \sqrt {x^{8} + 1}\right )}^{2} - 1}{3 \, {\left ({\left (x^{4} - \sqrt {x^{8} + 1}\right )}^{2} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 20, normalized size = 0.61 \[ \frac {\sqrt {x^{8}+1}\, \left (2 x^{8}-1\right )}{12 x^{12}} \]

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 = 1.08, size = 25, normalized size = 0.76 \[ \frac {\sqrt {x^{8} + 1}}{4 \, x^{4}} - \frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{12 \, x^{12}} \]

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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mupad [B]  time = 1.19, size = 19, normalized size = 0.58 \[ \frac {\sqrt {x^8+1}\,\left (2\,x^8-1\right )}{12\,x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^13*(x^8 + 1)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.57, size = 26, normalized size = 0.79 \[ \frac {\sqrt {1 + \frac {1}{x^{8}}}}{6} - \frac {\sqrt {1 + \frac {1}{x^{8}}}}{12 x^{8}} \]

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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