Integrand size = 13, antiderivative size = 87 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=-\frac {\sqrt [3]{1+x^2}}{2 x^2}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^2}\right )+\frac {1}{6} \log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]
-1/2*(x^2+1)^(1/3)/x^2+1/3*3^(1/2)*arctan(1/3*3^(1/2)+2/3*(x^2+1)^(1/3)*3^ (1/2))-1/3*ln(-1+(x^2+1)^(1/3))+1/6*ln(1+(x^2+1)^(1/3)+(x^2+1)^(2/3))
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{6} \left (-\frac {3 \sqrt [3]{1+x^2}}{x^2}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^2}}{\sqrt {3}}\right )-2 \log \left (-1+\sqrt [3]{1+x^2}\right )+\log \left (1+\sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right ) \]
((-3*(1 + x^2)^(1/3))/x^2 + 2*Sqrt[3]*ArcTan[(1 + 2*(1 + x^2)^(1/3))/Sqrt[ 3]] - 2*Log[-1 + (1 + x^2)^(1/3)] + Log[1 + (1 + x^2)^(1/3) + (1 + x^2)^(2 /3)])/6
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {243, 52, 69, 16, 1083, 217}
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.
\(\displaystyle \int \frac {1}{x^3 \left (x^2+1\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (x^2+1\right )^{2/3}}dx^2\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \int \frac {1}{x^2 \left (x^2+1\right )^{2/3}}dx^2-\frac {\sqrt [3]{x^2+1}}{x^2}\right )\) |
\(\Big \downarrow \) 69 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^2+1}}d\sqrt [3]{x^2+1}-\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{x^2+1}+1}d\sqrt [3]{x^2+1}-\frac {1}{2} \log \left (x^2\right )\right )-\frac {\sqrt [3]{x^2+1}}{x^2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{x^4+\sqrt [3]{x^2+1}+1}d\sqrt [3]{x^2+1}-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^2+1}\right )\right )-\frac {\sqrt [3]{x^2+1}}{x^2}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (3 \int \frac {1}{-x^4-3}d\left (2 \sqrt [3]{x^2+1}+1\right )-\frac {1}{2} \log \left (x^2\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^2+1}\right )\right )-\frac {\sqrt [3]{x^2+1}}{x^2}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )-\frac {\log \left (x^2\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^2+1}\right )\right )-\frac {\sqrt [3]{x^2+1}}{x^2}\right )\) |
(-((1 + x^2)^(1/3)/x^2) - (2*(-(Sqrt[3]*ArcTan[(1 + 2*(1 + x^2)^(1/3))/Sqr t[3]]) - Log[x^2]/2 + (3*Log[1 - (1 + x^2)^(1/3)])/2))/3)/2
3.12.77.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63
method | result | size |
meijerg | \(\frac {\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {8}{3}\right ], \left [2, 3\right ], -x^{2}\right )}{9}-\frac {2 \left (\frac {1}{2}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{2}}}{2 \Gamma \left (\frac {2}{3}\right )}\) | \(55\) |
risch | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {-\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{2}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}\) | \(59\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (1+2 \left (x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) x^{2}+\ln \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right ) x^{2}-2 \ln \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) x^{2}-3 \left (x^{2}+1\right )^{\frac {1}{3}}}{6 \left (-1+\left (x^{2}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{2}+1\right )^{\frac {1}{3}}+\left (x^{2}+1\right )^{\frac {2}{3}}\right )}\) | \(103\) |
trager | \(-\frac {\left (x^{2}+1\right )^{\frac {1}{3}}}{2 x^{2}}+\frac {\ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {2}{3}}+2 x^{2}+114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {1}{3}}+5}{x^{2}}\right )}{3}-2 \ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}+54 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}+90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {2}{3}}+2 x^{2}+114 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+9 \left (x^{2}+1\right )^{\frac {1}{3}}+5}{x^{2}}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+2 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}-90 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}-102 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-144 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2}-90 \left (x^{2}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+24 \left (x^{2}+1\right )^{\frac {2}{3}}+15 x^{2}-66 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+24 \left (x^{2}+1\right )^{\frac {1}{3}}+20}{x^{2}}\right )\) | \(441\) |
1/2/GAMMA(2/3)*(5/9*GAMMA(2/3)*x^2*hypergeom([1,1,8/3],[2,3],-x^2)-2/3*(1/ 2+1/6*Pi*3^(1/2)-3/2*ln(3)+2*ln(x))*GAMMA(2/3)-GAMMA(2/3)/x^2)
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{2} \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{6 \, x^{2}} \]
1/6*(2*sqrt(3)*x^2*arctan(2/3*sqrt(3)*(x^2 + 1)^(1/3) + 1/3*sqrt(3)) + x^2 *log((x^2 + 1)^(2/3) + (x^2 + 1)^(1/3) + 1) - 2*x^2*log((x^2 + 1)^(1/3) - 1) - 3*(x^2 + 1)^(1/3))/x^2
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=- \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 x^{\frac {10}{3}} \Gamma \left (\frac {8}{3}\right )} \]
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^2 + 1)^(1/3) + 1)) - 1/2*(x^2 + 1)^(1 /3)/x^2 + 1/6*log((x^2 + 1)^(2/3) + (x^2 + 1)^(1/3) + 1) - 1/3*log((x^2 + 1)^(1/3) - 1)
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{2} + 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {2}{3}} + {\left (x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^2 + 1)^(1/3) + 1)) - 1/2*(x^2 + 1)^(1 /3)/x^2 + 1/6*log((x^2 + 1)^(2/3) + (x^2 + 1)^(1/3) + 1) - 1/3*log((x^2 + 1)^(1/3) - 1)
Time = 5.54 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (1+x^2\right )^{2/3}} \, dx=-\frac {\ln \left ({\left (x^2+1\right )}^{1/3}-1\right )}{3}-\frac {{\left (x^2+1\right )}^{1/3}}{2\,x^2}-\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (3\,{\left (x^2+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]