\(\int \frac {1}{x^6 \sqrt {2+x^6}} \, dx\) [1401]

Optimal result

Integrand size = 13, antiderivative size = 181 \[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=-\frac {\sqrt {2+x^6}}{10 x^5}-\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]

[Out]

-1/10*(x^6+2)^(1/2)/x^5-1/60*x*(2^(1/3)+x^2)*((2^(1/3)+x^2*(1-3^(1/2)))^2/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)/(
2^(1/3)+x^2*(1-3^(1/2)))*(2^(1/3)+x^2*(1+3^(1/2)))*EllipticF((1-(2^(1/3)+x^2*(1-3^(1/2)))^2/(2^(1/3)+x^2*(1+3^
(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*((2^(2/3)-2^(1/3)*x^2+x^4)/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)*2^(2/3
)*3^(3/4)/(x^6+2)^(1/2)/(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*(1+3^(1/2)))^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 231} \[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=-\frac {x \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt {x^6+2}}{10 x^5} \]

[In]

Int[1/(x^6*Sqrt[2 + x^6]),x]

[Out]

-1/10*Sqrt[2 + x^6]/x^5 - (x*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3) + (1 + Sqrt[3])*x^2)^
2]*EllipticF[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(10*2^(1/3
)*3^(1/4)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6])

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 331

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2+x^6}}{10 x^5}-\frac {1}{5} \int \frac {1}{\sqrt {2+x^6}} \, dx \\ & = -\frac {\sqrt {2+x^6}}{10 x^5}-\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},-\frac {x^6}{2}\right )}{5 \sqrt {2} x^5} \]

[In]

Integrate[1/(x^6*Sqrt[2 + x^6]),x]

[Out]

-1/5*Hypergeometric2F1[-5/6, 1/2, 1/6, -1/2*x^6]/(Sqrt[2]*x^5)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 5.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.11

method	result	size
meijerg	\(-\frac {\sqrt {2}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};-\frac {x^{6}}{2}\right )}{10 x^{5}}\)	\(20\)
risch	\(-\frac {\sqrt {x^{6}+2}}{10 x^{5}}-\frac {\sqrt {2}\, x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\frac {x^{6}}{2}\right )}{10}\)	\(31\)

[In]

int(1/x^6/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/10*2^(1/2)/x^5*hypergeom([-5/6,1/2],[1/6],-1/2*x^6)

Fricas [F]

\[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x^6 + 2)/(x^12 + 2*x^6), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=\frac {\sqrt {2} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {1}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 x^{5} \Gamma \left (\frac {1}{6}\right )} \]

[In]

integrate(1/x**6/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*gamma(-5/6)*hyper((-5/6, 1/2), (1/6,), x**6*exp_polar(I*pi)/2)/(12*x**5*gamma(1/6))

Maxima [F]

\[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^6), x)

Giac [F]

\[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{6}} \,d x } \]

[In]

integrate(1/x^6/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \sqrt {2+x^6}} \, dx=\int \frac {1}{x^6\,\sqrt {x^6+2}} \,d x \]

[In]

int(1/(x^6*(x^6 + 2)^(1/2)),x)

[Out]

int(1/(x^6*(x^6 + 2)^(1/2)), x)