\(\int \frac {1}{x^3 \sqrt {2+x^6}} \, dx\) [1404]

Optimal result

Integrand size = 13, antiderivative size = 378 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {\sqrt {2+x^6}}{4 x^2}+\frac {\sqrt {2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {\left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]

[Out]

-1/4*(x^6+2)^(1/2)/x^2+1/4*(x^6+2)^(1/2)/(x^2+2^(1/3)*(1+3^(1/2)))+1/12*2^(2/3)*(2^(1/3)+x^2)*EllipticF((x^2+2
^(1/3)*(1-3^(1/2)))/(x^2+2^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+3^(1/2
)))^2)^(1/2)*3^(3/4)/(x^6+2)^(1/2)/((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+3^(1/2)))^2)^(1/2)-1/8*2^(1/6)*3^(1/4)*(2^(1
/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-3^(1/2)))/(x^2+2^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2)
)*((2^(2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+3^(1/2)))^2)^(1/2)/(x^6+2)^(1/2)/((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+3
^(1/2)))^2)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 331, 309, 224, 1891} \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\frac {\left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {\sqrt {x^6+2}}{4 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {\sqrt {x^6+2}}{4 x^2} \]

[In]

Int[1/(x^3*Sqrt[2 + x^6]),x]

[Out]

-1/4*Sqrt[2 + x^6]/x^2 + Sqrt[2 + x^6]/(4*(2^(1/3)*(1 + Sqrt[3]) + x^2)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(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[ArcSin[(2^(1/3)*(1 - Sqr
t[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(4*2^(5/6)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)^2]*Sqrt[2 + x^6]) + ((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[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(2*2^
(1/3)*3^(1/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 309

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

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 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {2+x^6}}{4 x^2}+\frac {1}{8} \text {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {2+x^6}}{4 x^2}+\frac {1}{8} \text {Subst}\left (\int \frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x}{\sqrt {2+x^3}} \, dx,x,x^2\right )-\frac {\left (1-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{4\ 2^{2/3}} \\ & = -\frac {\sqrt {2+x^6}}{4 x^2}+\frac {\sqrt {2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \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 (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {\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 (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\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 = 0.08 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},-\frac {x^6}{2}\right )}{2 \sqrt {2} x^2} \]

[In]

Integrate[1/(x^3*Sqrt[2 + x^6]),x]

[Out]

-1/2*Hypergeometric2F1[-1/3, 1/2, 2/3, -1/2*x^6]/(Sqrt[2]*x^2)

Maple [C] (verified)

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

Time = 5.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05

method	result	size
meijerg	\(-\frac {\sqrt {2}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};-\frac {x^{6}}{2}\right )}{4 x^{2}}\)	\(20\)
risch	\(-\frac {\sqrt {x^{6}+2}}{4 x^{2}}+\frac {\sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{32}\)	\(33\)

[In]

int(1/x^3/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {x^{2} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) + \sqrt {x^{6} + 2}}{4 \, x^{2}} \]

[In]

integrate(1/x^3/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

-1/4*(x^2*weierstrassZeta(0, -8, weierstrassPInverse(0, -8, x^2)) + sqrt(x^6 + 2))/x^2

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\frac {\sqrt {2} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 x^{2} \Gamma \left (\frac {2}{3}\right )} \]

[In]

integrate(1/x**3/(x**6+2)**(1/2),x)

[Out]

sqrt(2)*gamma(-1/3)*hyper((-1/3, 1/2), (2/3,), x**6*exp_polar(I*pi)/2)/(12*x**2*gamma(2/3))

Maxima [F]

\[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(x^6+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^3), x)

Giac [F]

\[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(x^6+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^6 + 2)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int \frac {1}{x^3\,\sqrt {x^6+2}} \,d x \]

[In]

int(1/(x^3*(x^6 + 2)^(1/2)),x)

[Out]

int(1/(x^3*(x^6 + 2)^(1/2)), x)