3.926 \(\int \frac{1}{x^{11} \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{4 \sqrt{x^4+1}}{15 x^2}+\frac{2 \sqrt{x^4+1}}{15 x^6}-\frac{\sqrt{x^4+1}}{10 x^{10}} \]

[Out]

-Sqrt[1 + x^4]/(10*x^10) + (2*Sqrt[1 + x^4])/(15*x^6) - (4*Sqrt[1 + x^4])/(15*x^2)

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Rubi [A]  time = 0.0098078, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ -\frac{4 \sqrt{x^4+1}}{15 x^2}+\frac{2 \sqrt{x^4+1}}{15 x^6}-\frac{\sqrt{x^4+1}}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^11*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(10*x^10) + (2*Sqrt[1 + x^4])/(15*x^6) - (4*Sqrt[1 + x^4])/(15*x^2)

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^{11} \sqrt{1+x^4}} \, dx &=-\frac{\sqrt{1+x^4}}{10 x^{10}}-\frac{4}{5} \int \frac{1}{x^7 \sqrt{1+x^4}} \, dx\\ &=-\frac{\sqrt{1+x^4}}{10 x^{10}}+\frac{2 \sqrt{1+x^4}}{15 x^6}+\frac{8}{15} \int \frac{1}{x^3 \sqrt{1+x^4}} \, dx\\ &=-\frac{\sqrt{1+x^4}}{10 x^{10}}+\frac{2 \sqrt{1+x^4}}{15 x^6}-\frac{4 \sqrt{1+x^4}}{15 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0048552, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^4+1} \left (8 x^8-4 x^4+3\right )}{30 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*Sqrt[1 + x^4]),x]

[Out]

-(Sqrt[1 + x^4]*(3 - 4*x^4 + 8*x^8))/(30*x^10)

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Maple [A]  time = 0.05, size = 25, normalized size = 0.5 \begin{align*} -{\frac{8\,{x}^{8}-4\,{x}^{4}+3}{30\,{x}^{10}}\sqrt{{x}^{4}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(x^4+1)^(1/2),x)

[Out]

-1/30*(x^4+1)^(1/2)*(8*x^8-4*x^4+3)/x^10

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Maxima [A]  time = 0.982142, size = 50, normalized size = 1.02 \begin{align*} -\frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} + \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{3 \, x^{6}} - \frac{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}{10 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(x^4 + 1)/x^2 + 1/3*(x^4 + 1)^(3/2)/x^6 - 1/10*(x^4 + 1)^(5/2)/x^10

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Fricas [A]  time = 1.41508, size = 77, normalized size = 1.57 \begin{align*} -\frac{8 \, x^{10} +{\left (8 \, x^{8} - 4 \, x^{4} + 3\right )} \sqrt{x^{4} + 1}}{30 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/30*(8*x^10 + (8*x^8 - 4*x^4 + 3)*sqrt(x^4 + 1))/x^10

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Sympy [A]  time = 2.08609, size = 44, normalized size = 0.9 \begin{align*} - \frac{4 \sqrt{1 + \frac{1}{x^{4}}}}{15} + \frac{2 \sqrt{1 + \frac{1}{x^{4}}}}{15 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{10 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(x**4+1)**(1/2),x)

[Out]

-4*sqrt(1 + x**(-4))/15 + 2*sqrt(1 + x**(-4))/(15*x**4) - sqrt(1 + x**(-4))/(10*x**8)

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Giac [A]  time = 1.21951, size = 38, normalized size = 0.78 \begin{align*} -\frac{1}{10} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

-1/10*(1/x^4 + 1)^(5/2) + 1/3*(1/x^4 + 1)^(3/2) - 1/2*sqrt(1/x^4 + 1)