Integrand size = 16, antiderivative size = 95 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=\frac {(2-3 x) \left (1+x^3\right )^{2/3}}{6 x^3}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{18} \log \left (1+\sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \] Output:
1/6*(2-3*x)*(x^3+1)^(2/3)/x^3+1/9*arctan(1/3*3^(1/2)+2/3*(x^3+1)^(1/3)*3^( 1/2))*3^(1/2)+1/9*ln(-1+(x^3+1)^(1/3))-1/18*ln(1+(x^3+1)^(1/3)+(x^3+1)^(2/ 3))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.37 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=-\frac {\left (1+x^3\right )^{2/3} \left (1+x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},1+x^3\right )\right )}{2 x^2} \] Input:
Integrate[(-1 + x)/(x^4*(1 + x^3)^(1/3)),x]
Output:
-1/2*((1 + x^3)^(2/3)*(1 + x^2*Hypergeometric2F1[2/3, 2, 5/3, 1 + x^3]))/x ^2
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2383, 2009}
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.
\(\displaystyle \int \frac {x-1}{x^4 \sqrt [3]{x^3+1}} \, dx\) |
\(\Big \downarrow \) 2383 |
\(\displaystyle \int \left (\frac {1}{x^3 \sqrt [3]{x^3+1}}-\frac {1}{x^4 \sqrt [3]{x^3+1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{3 x^3}+\frac {1}{6} \log \left (1-\sqrt [3]{x^3+1}\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {\log (x)}{6}\) |
Input:
Int[(-1 + x)/(x^4*(1 + x^3)^(1/3)),x]
Output:
(1 + x^3)^(2/3)/(3*x^3) - (1 + x^3)^(2/3)/(2*x^2) + ArcTan[(1 + 2*(1 + x^3 )^(1/3))/Sqrt[3]]/(3*Sqrt[3]) - Log[x]/6 + Log[1 - (1 + x^3)^(1/3)]/6
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {4 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(90\) |
risch | \(-\frac {3 x^{4}-2 x^{3}+3 x -2}{6 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi }\) | \(91\) |
trager | \(-\frac {\left (-2+3 x \right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{3}}+\frac {\ln \left (-\frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+6 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-30 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-10 x^{3}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+48 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-20 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+15 \left (x^{3}+1\right )^{\frac {1}{3}}-25}{x^{3}}\right )}{9}+\frac {2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {20 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+32 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-30 \left (x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+3 x^{3}-20 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-18 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}-24 \left (x^{3}+1\right )^{\frac {2}{3}}+38 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+15 \left (x^{3}+1\right )^{\frac {1}{3}}+4}{x^{3}}\right )}{9}\) | \(299\) |
Input:
int((-1+x)/x^4/(x^3+1)^(1/3),x,method=_RETURNVERBOSE)
Output:
-1/6/Pi*3^(1/2)*GAMMA(2/3)*(4/27*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1, 7/3],[2,3],-x^3)-2/9*(2-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x))*Pi*3^(1/2)/GAMMA (2/3)-2/3*Pi*3^(1/2)/GAMMA(2/3)/x^3)-1/2*(x^3+1)^(2/3)/x^2
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=-\frac {2 \, \sqrt {3} x^{3} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - x^{3} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (3 \, x - 2\right )}}{18 \, x^{3}} \] Input:
integrate((-1+x)/x^4/(x^3+1)^(1/3),x, algorithm="fricas")
Output:
-1/18*(2*sqrt(3)*x^3*arctan(-(sqrt(3)*(x^3 + 1) - 2*sqrt(3)*(x^3 + 1)^(2/3 ) + 4*sqrt(3)*(x^3 + 1)^(1/3))/(x^3 + 9)) - x^3*log((x^3 - 3*(x^3 + 1)^(2/ 3) + 3*(x^3 + 1)^(1/3))/x^3) + 3*(x^3 + 1)^(2/3)*(3*x - 2))/x^3
Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.56 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=\frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {2}{3}\right )}{3 \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{4} \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((-1+x)/x**4/(x**3+1)**(1/3),x)
Output:
(1 + x**(-3))**(2/3)*gamma(-2/3)/(3*gamma(1/3)) + gamma(4/3)*hyper((1/3, 4 /3), (7/3,), exp_polar(I*pi)/x**3)/(3*x**4*gamma(7/3))
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \] Input:
integrate((-1+x)/x^4/(x^3+1)^(1/3),x, algorithm="maxima")
Output:
1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 + 1)^(1/3) + 1)) - 1/2*(x^3 + 1)^(2 /3)/x^2 + 1/3*(x^3 + 1)^(2/3)/x^3 - 1/18*log((x^3 + 1)^(2/3) + (x^3 + 1)^( 1/3) + 1) + 1/9*log((x^3 + 1)^(1/3) - 1)
\[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=\int { \frac {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{4}} \,d x } \] Input:
integrate((-1+x)/x^4/(x^3+1)^(1/3),x, algorithm="giac")
Output:
integrate((x - 1)/((x^3 + 1)^(1/3)*x^4), x)
Time = 8.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09 \[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=\frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-9\,{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3+1\right )}^{2/3}}{2\,x^2}+\frac {{\left (x^3+1\right )}^{2/3}}{3\,x^3} \] Input:
int((x - 1)/(x^4*(x^3 + 1)^(1/3)),x)
Output:
log((x^3 + 1)^(1/3)/9 - 1/9)/9 + log((x^3 + 1)^(1/3)/9 - 9*((3^(1/2)*1i)/1 8 - 1/18)^2)*((3^(1/2)*1i)/18 - 1/18) - log((x^3 + 1)^(1/3)/9 - 9*((3^(1/2 )*1i)/18 + 1/18)^2)*((3^(1/2)*1i)/18 + 1/18) - (x^3 + 1)^(2/3)/(2*x^2) + ( x^3 + 1)^(2/3)/(3*x^3)
\[ \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx=-\left (\int \frac {1}{\left (x^{3}+1\right )^{\frac {1}{3}} x^{4}}d x \right )+\int \frac {1}{\left (x^{3}+1\right )^{\frac {1}{3}} x^{3}}d x \] Input:
int((-1+x)/x^4/(x^3+1)^(1/3),x)
Output:
- int(1/((x**3 + 1)**(1/3)*x**4),x) + int(1/((x**3 + 1)**(1/3)*x**3),x)