Integrand size = 40, antiderivative size = 230 \[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^2-x^3}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^2-x^3}}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1+x^2-x^3}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1+x^2-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2-x^3}+\left (1+x^2-x^3\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^2-x^3}-\sqrt [3]{2} \left (1+x^2-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
3^(1/2)*arctan(3^(1/2)*x/(-x+2*(-x^3+x^2+1)^(1/3)))-1/2*3^(1/2)*arctan(3^( 1/2)*x/(-x+2^(2/3)*(-x^3+x^2+1)^(1/3)))*2^(2/3)+ln(x+(-x^3+x^2+1)^(1/3))-1 /2*ln(2*x+2^(2/3)*(-x^3+x^2+1)^(1/3))*2^(2/3)-1/2*ln(x^2-x*(-x^3+x^2+1)^(1 /3)+(-x^3+x^2+1)^(2/3))+1/4*ln(-2*x^2+2^(2/3)*x*(-x^3+x^2+1)^(1/3)-2^(1/3) *(-x^3+x^2+1)^(2/3))*2^(2/3)
Time = 0.61 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^2-x^3}}\right )-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^2-x^3}}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1+x^2-x^3}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1+x^2-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2-x^3}+\left (1+x^2-x^3\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^2-x^3}-\sqrt [3]{2} \left (1+x^2-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^2 - x^3)^(1/3))] - (Sqrt[3]*ArcT an[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^2 - x^3)^(1/3))])/2^(1/3) + Log[x + (1 + x^2 - x^3)^(1/3)] - Log[2*x + 2^(2/3)*(1 + x^2 - x^3)^(1/3)]/2^(1/3) - Log[x^2 - x*(1 + x^2 - x^3)^(1/3) + (1 + x^2 - x^3)^(2/3)]/2 + Log[-2*x^2 + 2^(2/3)*x*(1 + x^2 - x^3)^(1/3) - 2^(1/3)*(1 + x^2 - x^3)^(2/3)]/(2*2^(1 /3))
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.
| \(\displaystyle \int \frac {x^3 \left (x^2+3\right )}{\left (x^2+1\right ) \sqrt [3]{-x^3+x^2+1} \left (x^3+x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {-x^2-3}{\sqrt [3]{-x^3+x^2+1} \left (x^3+x^2+1\right )}+\frac {2}{\left (x^2+1\right ) \sqrt [3]{-x^3+x^2+1}}+\frac {1}{\sqrt [3]{-x^3+x^2+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{-6 x+\sqrt [3]{2} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}\right )+2} \left (-6 x+\frac {2 \sqrt [3]{2}+\left (58+6 \sqrt {93}\right )^{2/3}}{\sqrt [3]{29+3 \sqrt {93}}}+2\right )^{2/3} \sqrt [3]{1-\frac {2^{2/3} \left (-6 \sqrt [3]{29+3 \sqrt {93}} x+\left (58+6 \sqrt {93}\right )^{2/3}+2 \sqrt [3]{29+3 \sqrt {93}}+2 \sqrt [3]{2}\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1-\frac {2^{2/3} \left (-6 \sqrt [3]{29+3 \sqrt {93}} x+\left (58+6 \sqrt {93}\right )^{2/3}+2 \sqrt [3]{29+3 \sqrt {93}}+2 \sqrt [3]{2}\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (2^{2/3} (1-3 x)+\sqrt [3]{58+6 \sqrt {93}}+\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}},\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (2^{2/3} (1-3 x)+\sqrt [3]{58+6 \sqrt {93}}+\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}\right )}{4 \sqrt [3]{-x^3+x^2+1}}+\frac {i \sqrt [3]{2 (1-3 x)^2-2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (1-3 x)+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-2} \sqrt [3]{-6 x+\sqrt [3]{2} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}\right )+2} \text {Subst}\left (\int \frac {1}{\left (-x-\left (\frac {1}{3}-i\right )\right ) \sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \sqrt [3]{x^2+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}dx,x,x-\frac {1}{3}\right )}{3\ 2^{2/3} \sqrt [3]{-x^3+x^2+1}}+\frac {i \sqrt [3]{2 (1-3 x)^2-2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (1-3 x)+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-2} \sqrt [3]{-6 x+\sqrt [3]{2} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}\right )+2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \left (x+\left (\frac {1}{3}+i\right )\right ) \sqrt [3]{x^2+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}dx,x,x-\frac {1}{3}\right )}{3\ 2^{2/3} \sqrt [3]{-x^3+x^2+1}}-3 \int \frac {1}{\sqrt [3]{-x^3+x^2+1} \left (x^3+x^2+1\right )}dx-\int \frac {x^2}{\sqrt [3]{-x^3+x^2+1} \left (x^3+x^2+1\right )}dx\) |
3.27.22.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.94 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\ln \left (\frac {x +\left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}-x \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{3}+x^{2}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}}{x}\right )}{2}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}} x +\left (-x^{3}+x^{2}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{4}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )}{2}\) | \(200\) |
| trager | \(\text {Expression too large to display}\) | \(1216\) |
ln((x+(-x^3+x^2+1)^(1/3))/x)-1/2*ln((x^2-x*(-x^3+x^2+1)^(1/3)+(-x^3+x^2+1) ^(2/3))/x^2)+3^(1/2)*arctan(1/3*(x-2*(-x^3+x^2+1)^(1/3))*3^(1/2)/x)-1/2*2^ (2/3)*ln((2^(1/3)*x+(-x^3+x^2+1)^(1/3))/x)+1/4*2^(2/3)*ln((2^(2/3)*x^2-2^( 1/3)*(-x^3+x^2+1)^(1/3)*x+(-x^3+x^2+1)^(2/3))/x^2)-1/2*3^(1/2)*2^(2/3)*arc tan(1/3*3^(1/2)*(x-2^(2/3)*(-x^3+x^2+1)^(1/3))/x)
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.05 \[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\frac {1}{4} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} \log \left (-\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} - 6 \cdot 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, x - 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} - 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \log \left (\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} + 32 \, x + 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} - {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
1/4*2^(2/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))*log(-1/8*(x*(I*sqrt(3)*(-1 )^(1/3) - (-1)^(1/3))^3 - 6*2^(1/3)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^ 2 + 8*x - 24*(-x^3 + x^2 + 1)^(1/3))/x) - 1/8*(2^(2/3)*(I*sqrt(3)*(-1)^(1/ 3) - (-1)^(1/3)) - 2*sqrt(3/2)*sqrt(-2^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^ (1/3))^2))*log(-3/8*(2^(2/3)*sqrt(3/2)*sqrt(-2^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3)) + 2^(1/3)*x*(I*sqr t(3)*(-1)^(1/3) - (-1)^(1/3))^2 - 8*(-x^3 + x^2 + 1)^(1/3))/x) - 1/8*(2^(2 /3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3)) + 2*sqrt(3/2)*sqrt(-2^(1/3)*(I*sqr t(3)*(-1)^(1/3) - (-1)^(1/3))^2))*log(3/8*(2^(2/3)*sqrt(3/2)*sqrt(-2^(1/3) *(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/ 3)) - 2^(1/3)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2 + 8*(-x^3 + x^2 + 1) ^(1/3))/x) - sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + x^2 + 1)^( 1/3))/x) + log(1/8*(x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^3 + 32*x + 24*(- x^3 + x^2 + 1)^(1/3))/x) - 1/2*log((x^2 - (-x^3 + x^2 + 1)^(1/3)*x + (-x^3 + x^2 + 1)^(2/3))/x^2)
\[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\int \frac {x^{3} \left (x^{2} + 3\right )}{\left (x^{2} + 1\right ) \sqrt [3]{- x^{3} + x^{2} + 1} \left (x^{3} + x^{2} + 1\right )}\, dx \]
\[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{2} + 3\right )} x^{3}}{{\left (x^{3} + x^{2} + 1\right )} {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}} \,d x } \]
\[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{2} + 3\right )} x^{3}}{{\left (x^{3} + x^{2} + 1\right )} {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx=\int \frac {x^3\,\left (x^2+3\right )}{\left (x^2+1\right )\,\left (x^3+x^2+1\right )\,{\left (-x^3+x^2+1\right )}^{1/3}} \,d x \]