Integrand size = 27, antiderivative size = 44 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x \sqrt {1-x^2+x^3}}{1+x^3}-\arctan \left (\frac {x}{\sqrt {1-x^2+x^3}}\right ) \]
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x \sqrt {1-x^2+x^3}}{1+x^3}-\arctan \left (\frac {x}{\sqrt {1-x^2+x^3}}\right ) \]
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.
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+1}}{x^3+1}-\frac {3 \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (x^3-2\right ) \sqrt {x^3-x^2+1}}{\left (x^3+1\right )^2}dx\) |
3.6.68.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {x \sqrt {x^{3}-x^{2}+1}}{x^{3}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )\) | \(41\) |
default | \(\frac {\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right ) x^{3}-x \sqrt {x^{3}-x^{2}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )}{x^{3}+1}\) | \(63\) |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right ) x^{3}-x \sqrt {x^{3}-x^{2}+1}+\arctan \left (\frac {\sqrt {x^{3}-x^{2}+1}}{x}\right )}{x^{3}+1}\) | \(63\) |
trager | \(-\frac {x \sqrt {x^{3}-x^{2}+1}}{x^{3}+1}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}-x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{2}\) | \(93\) |
elliptic | \(\text {Expression too large to display}\) | \(50736\) |
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=\frac {{\left (x^{3} + 1\right )} \arctan \left (\frac {\sqrt {x^{3} - x^{2} + 1} {\left (x^{3} - 2 \, x^{2} + 1\right )}}{2 \, {\left (x^{4} - x^{3} + x\right )}}\right ) - 2 \, \sqrt {x^{3} - x^{2} + 1} x}{2 \, {\left (x^{3} + 1\right )}} \]
1/2*((x^3 + 1)*arctan(1/2*sqrt(x^3 - x^2 + 1)*(x^3 - 2*x^2 + 1)/(x^4 - x^3 + x)) - 2*sqrt(x^3 - x^2 + 1)*x)/(x^3 + 1)
\[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=\int \frac {\left (x^{3} - 2\right ) \sqrt {x^{3} - x^{2} + 1}}{\left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2}}\, dx \]
\[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 1} {\left (x^{3} - 2\right )}}{{\left (x^{3} + 1\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 1} {\left (x^{3} - 2\right )}}{{\left (x^{3} + 1\right )}^{2}} \,d x } \]
Time = 6.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx=-\frac {x\,\sqrt {x^3-x^2+1}}{x^3+1}+\frac {\ln \left (\frac {x^3-2\,x^2+1+x\,\sqrt {x^3-x^2+1}\,2{}\mathrm {i}}{x^3+1}\right )\,1{}\mathrm {i}}{2} \]