\(\int \frac {1}{x^2 \sqrt {2+x^6}} \, dx\) [1407]

Optimal result

Integrand size = 13, antiderivative size = 392 \[ \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx=-\frac {\sqrt {2+x^6}}{2 x}+\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{2 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}-\frac {\sqrt [4]{3} x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}-\frac {\left (1-\sqrt {3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]

[Out]

-1/2*(x^6+2)^(1/2)/x+1/2*x*(1+3^(1/2))*(x^6+2)^(1/2)/(2^(1/3)+x^2*(1+3^(1/2)))-1/2*3^(1/4)*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)
))*EllipticE((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^(1/3)/(x^6+2)^(1/2)/(x^2*(2^(1/3)+x^2)/(2^(1/3)+x^2*
(1+3^(1/2)))^2)^(1/2)-1/12*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))*(1-3^(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^(1/3)*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 = 0.07 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {331, 314, 231, 1895} \[ \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx=-\frac {\left (1-\sqrt {3}\right ) \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [4]{3} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} x E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2^{2/3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt {x^6+2}}{2 x}+\frac {\left (1+\sqrt {3}\right ) \sqrt {x^6+2} x}{2 \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )} \]

[In]

Int[1/(x^2*Sqrt[2 + x^6]),x]

[Out]

-1/2*Sqrt[2 + x^6]/x + ((1 + Sqrt[3])*x*Sqrt[2 + x^6])/(2*(2^(1/3) + (1 + Sqrt[3])*x^2)) - (3^(1/4)*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]*EllipticE[ArcCos[(2^(1/3) + (1 - S
qrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(2^(2/3)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) +
(1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6]) - ((1 - Sqrt[3])*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])/(2*2^(2/3)*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 314

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

Rule 1895

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/
a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqrt[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d
*s*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*r^2*Sqrt[(r*x^2*(s + r*x^2))/
(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]))*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r
*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2+x^6}}{2 x}+\int \frac {x^4}{\sqrt {2+x^6}} \, dx \\ & = -\frac {\sqrt {2+x^6}}{2 x}-\frac {1}{2} \int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2+x^6}} \, dx-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {2+x^6}} \, dx}{\sqrt [3]{2}} \\ & = -\frac {\sqrt {2+x^6}}{2 x}+\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{2 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}-\frac {\sqrt [4]{3} 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}} E\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 )}{2^{2/3} \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}}-\frac {\left (1-\sqrt {3}\right ) 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 )}{2\ 2^{2/3} \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 = 27, normalized size of antiderivative = 0.07 \[ \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},-\frac {x^6}{2}\right )}{\sqrt {2} x} \]

[In]

Integrate[1/(x^2*Sqrt[2 + x^6]),x]

[Out]

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

Maple [C] (verified)

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

Time = 5.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05

method	result	size
meijerg	\(-\frac {\sqrt {2}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};-\frac {x^{6}}{2}\right )}{2 x}\)	\(20\)
risch	\(-\frac {\sqrt {x^{6}+2}}{2 x}+\frac {\sqrt {2}\, x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\frac {x^{6}}{2}\right )}{10}\)	\(33\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(x^6 + 2)/(x^8 + 2*x^2), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx=\frac {\sqrt {2} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 x \Gamma \left (\frac {5}{6}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/(sqrt(x^6 + 2)*x^2), x)

Mupad [B] (verification not implemented)

Time = 5.89 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^2 \sqrt {2+x^6}} \, dx=-\frac {\sqrt {\frac {2}{x^6}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ -\frac {2}{x^6}\right )}{4\,x\,\sqrt {x^6+2}} \]

[In]

int(1/(x^2*(x^6 + 2)^(1/2)),x)

[Out]

-((2/x^6 + 1)^(1/2)*hypergeom([1/2, 2/3], 5/3, -2/x^6))/(4*x*(x^6 + 2)^(1/2))