Integrand size = 18, antiderivative size = 97 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {\sqrt [3]{1+x^3} \left (1+3 x^3\right )}{3 x^3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
1/3*(x^3+1)^(1/3)*(3*x^3+1)/x^3-2/9*arctan(1/3*3^(1/2)+2/3*(x^3+1)^(1/3)*3 ^(1/2))*3^(1/2)+2/9*ln(-1+(x^3+1)^(1/3))-1/9*ln(1+(x^3+1)^(1/3)+(x^3+1)^(2 /3))
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {1}{9} \left (\frac {3 \sqrt [3]{1+x^3} \left (1+3 x^3\right )}{x^3}-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^3}\right )-\log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )\right ) \]
((3*(1 + x^3)^(1/3)*(1 + 3*x^3))/x^3 - 2*Sqrt[3]*ArcTan[(1 + 2*(1 + x^3)^( 1/3))/Sqrt[3]] + 2*Log[-1 + (1 + x^3)^(1/3)] - Log[1 + (1 + x^3)^(1/3) + ( 1 + x^3)^(2/3)])/9
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {948, 25, 87, 60, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt [3]{x^3+1}}{x^4} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int -\frac {\left (1-x^3\right ) \sqrt [3]{x^3+1}}{x^6}dx^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {\left (1-x^3\right ) \sqrt [3]{x^3+1}}{x^6}dx^3\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \int \frac {\sqrt [3]{x^3+1}}{x^3}dx^3+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (\int \frac {1}{x^3 \left (x^3+1\right )^{2/3}}dx^3+3 \sqrt [3]{x^3+1}\right )+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{1-\sqrt [3]{x^3+1}}d\sqrt [3]{x^3+1}-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{x^3+1}+1}d\sqrt [3]{x^3+1}+3 \sqrt [3]{x^3+1}-\frac {1}{2} \log \left (x^3\right )\right )+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+\sqrt [3]{x^3+1}+1}d\sqrt [3]{x^3+1}+3 \sqrt [3]{x^3+1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3+1}+1\right )+3 \sqrt [3]{x^3+1}-\frac {1}{2} \log \left (x^3\right )+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )+3 \sqrt [3]{x^3+1}-\frac {\log \left (x^3\right )}{2}+\frac {3}{2} \log \left (1-\sqrt [3]{x^3+1}\right )\right )+\frac {\left (x^3+1\right )^{4/3}}{x^3}\right )\) |
((1 + x^3)^(4/3)/x^3 + (2*(3*(1 + x^3)^(1/3) - Sqrt[3]*ArcTan[(1 + 2*(1 + x^3)^(1/3))/Sqrt[3]] - Log[x^3]/2 + (3*Log[1 - (1 + x^3)^(1/3)])/2))/3)/3
3.14.43.3.1 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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
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 = 2.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {3 x^{6}+4 x^{3}+1}{3 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}}+\frac {-\frac {4 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{27}+\frac {2 \left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{9}}{\Gamma \left (\frac {2}{3}\right )}\) | \(71\) |
| meijerg | \(\frac {\frac {\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{3}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{3}}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\) | \(103\) |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{3}+9 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}+2 \ln \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) x^{3}-\ln \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right ) x^{3}+3 \left (x^{3}+1\right )^{\frac {1}{3}}}{9 \left (-1+\left (x^{3}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}\right )}\) | \(116\) |
| trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (3 x^{3}+1\right )}{3 x^{3}}+\frac {2 \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-12 x^{3}+24 \left (x^{3}+1\right )^{\frac {2}{3}}-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9 \left (x^{3}+1\right )^{\frac {1}{3}}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-16}{x^{3}}\right )}{9}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-8 x^{3}-9 \left (x^{3}+1\right )^{\frac {2}{3}}+9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+24 \left (x^{3}+1\right )^{\frac {1}{3}}-29 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-20}{x^{3}}\right )}{9}\) | \(248\) |
1/3*(3*x^6+4*x^3+1)/x^3/(x^3+1)^(2/3)+2/9/GAMMA(2/3)*(-2/3*GAMMA(2/3)*x^3* hypergeom([1,1,5/3],[2,2],-x^3)+(1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*GAMMA(2 /3))
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{3}} \]
-1/9*(2*sqrt(3)*x^3*arctan(2/3*sqrt(3)*(x^3 + 1)^(1/3) + 1/3*sqrt(3)) + x^ 3*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1) - 2*x^3*log((x^3 + 1)^(1/3) - 1) - 3*(3*x^3 + 1)*(x^3 + 1)^(1/3))/x^3
Result contains complex when optimal does not.
Time = 51.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=- \frac {x \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 \Gamma \left (\frac {2}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \]
-x*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), exp_polar(I*pi)/x**3)/(3*gamma( 2/3)) + gamma(2/3)*hyper((-1/3, 2/3), (5/3,), exp_polar(I*pi)/x**3)/(3*x** 2*gamma(5/3))
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + (x^3 + 1)^(1/3) + 1/3*(x^3 + 1)^(1/3)/x^3 - 1/9*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1 ) + 2/9*log((x^3 + 1)^(1/3) - 1)
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} - \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{9} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) + (x^3 + 1)^(1/3) + 1/3*(x^3 + 1)^(1/3)/x^3 - 1/9*log((x^3 + 1)^(2/3) + (x^3 + 1)^(1/3) + 1 ) + 2/9*log(abs((x^3 + 1)^(1/3) - 1))
Time = 6.82 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^4} \, dx=\frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+{\left (x^3+1\right )}^{1/3}-\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left ({\left (x^3+1\right )}^{1/3}+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {{\left (x^3+1\right )}^{1/3}}{3\,x^3}+\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
log((x^3 + 1)^(1/3) - 1)/3 - log((x^3 + 1)^(1/3)/9 - 1/9)/9 + (x^3 + 1)^(1 /3) - log((x^3 + 1)^(1/3) - (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*1i)/18 - 1/18) + log((3^(1/2)*1i)/2 + (x^3 + 1)^(1/3) + 1/2)*((3^(1/2)*1i)/18 + 1/18) + (x^3 + 1)^(1/3)/(3*x^3) + log(3*(x^3 + 1)^(1/3) - (3^(1/2)*3i)/2 + 3/2)*(( 3^(1/2)*1i)/6 - 1/6) - log((3^(1/2)*3i)/2 + 3*(x^3 + 1)^(1/3) + 3/2)*((3^( 1/2)*1i)/6 + 1/6)