3.925 \(\int \frac{1}{x^7 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\sqrt{x^4+1}}{3 x^2}-\frac{\sqrt{x^4+1}}{6 x^6} \]

[Out]

-Sqrt[1 + x^4]/(6*x^6) + Sqrt[1 + x^4]/(3*x^2)

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Rubi [A]  time = 0.005913, 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} \[ \frac{\sqrt{x^4+1}}{3 x^2}-\frac{\sqrt{x^4+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(6*x^6) + Sqrt[1 + x^4]/(3*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^7 \sqrt{1+x^4}} \, dx &=-\frac{\sqrt{1+x^4}}{6 x^6}-\frac{2}{3} \int \frac{1}{x^3 \sqrt{1+x^4}} \, dx\\ &=-\frac{\sqrt{1+x^4}}{6 x^6}+\frac{\sqrt{1+x^4}}{3 x^2}\\ \end{align*}

Mathematica [A]  time = 0.003738, size = 23, normalized size = 0.7 \[ -\frac{\left (1-2 x^4\right ) \sqrt{x^4+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

-((1 - 2*x^4)*Sqrt[1 + x^4])/(6*x^6)

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Maple [A]  time = 0.07, size = 20, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{4}-1}{6\,{x}^{6}}\sqrt{{x}^{4}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.01636, size = 34, normalized size = 1.03 \begin{align*} \frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45533, size = 61, normalized size = 1.85 \begin{align*} \frac{2 \, x^{6} +{\left (2 \, x^{4} - 1\right )} \sqrt{x^{4} + 1}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^4+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.10802, size = 26, normalized size = 0.79 \begin{align*} \frac{\sqrt{1 + \frac{1}{x^{4}}}}{3} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**4+1)**(1/2),x)

[Out]

sqrt(1 + x**(-4))/3 - sqrt(1 + x**(-4))/(6*x**4)

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Giac [A]  time = 1.19285, size = 26, normalized size = 0.79 \begin{align*} -\frac{1}{6} \,{\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^7/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

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