Integrand size = 13, antiderivative size = 408 \[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=\frac {1}{6 x \sqrt {2+x^6}}-\frac {\sqrt {2+x^6}}{3 x}+\frac {\left (1+\sqrt {3}\right ) x \sqrt {2+x^6}}{3 \left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )}-\frac {\sqrt [3]{2} x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}}-\frac {\left (1-\sqrt {3}\right ) x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]
1/6/x/(x^6+2)^(1/2)-1/3*(x^6+2)^(1/2)/x+1/3*x*(1+3^(1/2))*(x^6+2)^(1/2)/(2 ^(1/3)+x^2*(1+3^(1/2)))-1/3*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)-1/18*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)
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 = 0.07 \[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{2},\frac {5}{6},-\frac {x^6}{2}\right )}{2 \sqrt {2} x} \]
Time = 0.47 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {819, 847, 837, 25, 27, 766, 2420}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (x^6+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {2}{3} \int \frac {1}{x^2 \sqrt {x^6+2}}dx+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {2}{3} \left (\int \frac {x^4}{\sqrt {x^6+2}}dx-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {2}{3} \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^6+2}}dx}{\sqrt [3]{2}}-\frac {1}{2} \int -\frac {2 x^4+2^{2/3} \left (1-\sqrt {3}\right )}{\sqrt {x^6+2}}dx-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^6+2}}dx}{\sqrt [3]{2}}+\frac {1}{2} \int \frac {2^{2/3} \left (\sqrt [3]{2} x^4-\sqrt {3}+1\right )}{\sqrt {x^6+2}}dx-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} \left (-\frac {\left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^6+2}}dx}{\sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{2} x^4-\sqrt {3}+1}{\sqrt {x^6+2}}dx}{\sqrt [3]{2}}-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {2}{3} \left (\frac {\int \frac {\sqrt [3]{2} x^4-\sqrt {3}+1}{\sqrt {x^6+2}}dx}{\sqrt [3]{2}}-\frac {\left (1-\sqrt {3}\right ) x \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {2}{3} \left (-\frac {\left (1-\sqrt {3}\right ) x \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2\ 2^{2/3} \sqrt [4]{3} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}+\frac {\frac {\left (1+\sqrt {3}\right ) x \sqrt {x^6+2}}{2^{2/3} \left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )}-\frac {\sqrt [4]{3} x \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [3]{2} \sqrt {\frac {x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt {3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt {x^6+2}}}{\sqrt [3]{2}}-\frac {\sqrt {x^6+2}}{2 x}\right )+\frac {1}{6 x \sqrt {x^6+2}}\) |
1/(6*x*Sqrt[2 + x^6]) + (2*(-1/2*Sqrt[2 + x^6]/x + (((1 + Sqrt[3])*x*Sqrt[ 2 + x^6])/(2^(2/3)*(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]*E llipticE[ArcCos[(2^(1/3) + (1 - Sqrt[3])*x^2)/(2^(1/3) + (1 + Sqrt[3])*x^2 )], (2 + Sqrt[3])/4])/(2^(1/3)*Sqrt[(x^2*(2^(1/3) + x^2))/(2^(1/3) + (1 + Sqrt[3])*x^2)^2]*Sqrt[2 + x^6]))/2^(1/3) - ((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]*Elli pticF[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])))/3
3.15.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
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] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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*(Sqr t[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]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
| method | result | size |
| meijerg | \(-\frac {\sqrt {2}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{6},\frac {3}{2};\frac {5}{6};-\frac {x^{6}}{2}\right )}{4 x}\) | \(20\) |
| risch | \(-\frac {2 x^{6}+3}{6 x \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, x^{5} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\frac {x^{6}}{2}\right )}{15}\) | \(40\) |
\[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {2} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 x \Gamma \left (\frac {5}{6}\right )} \]
\[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=\int { \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Time = 6.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^2 \left (2+x^6\right )^{3/2}} \, dx=-\frac {{\left (\frac {2}{x^6}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{3};\ \frac {8}{3};\ -\frac {2}{x^6}\right )}{10\,x\,{\left (x^6+2\right )}^{3/2}} \]