Integrand size = 30, antiderivative size = 68 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=\frac {\sqrt {-1+x^3} \left (-2+5 x^3-12 x^6\right )}{36 x^9}+\frac {13}{24} \arctan \left (\sqrt {-1+x^3}\right )-\frac {7 \arctan \left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}} \]
1/36*(x^3-1)^(1/2)*(-12*x^6+5*x^3-2)/x^9+13/24*arctan((x^3-1)^(1/2))-7/24* arctan(1/3*(x^3-1)^(1/2)*3^(1/2))*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=\frac {\sqrt {-1+x^3} \left (-2+5 x^3-12 x^6\right )}{36 x^9}+\frac {13}{24} \arctan \left (\sqrt {-1+x^3}\right )-\frac {7 \arctan \left (\frac {\sqrt {-1+x^3}}{\sqrt {3}}\right )}{8 \sqrt {3}} \]
(Sqrt[-1 + x^3]*(-2 + 5*x^3 - 12*x^6))/(36*x^9) + (13*ArcTan[Sqrt[-1 + x^3 ]])/24 - (7*ArcTan[Sqrt[-1 + x^3]/Sqrt[3]])/(8*Sqrt[3])
Time = 0.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7276, 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.
\(\displaystyle \int \frac {\sqrt {x^3-1} \left (x^6-x^3+1\right )}{x^{10} \left (x^3+2\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {7 \sqrt {x^3-1}}{16 x}+\frac {\sqrt {x^3-1}}{2 x^{10}}-\frac {3 \sqrt {x^3-1}}{4 x^7}+\frac {7 \sqrt {x^3-1}}{8 x^4}+\frac {7 \sqrt {x^3-1} x^2}{16 \left (x^3+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {13}{24} \arctan \left (\sqrt {x^3-1}\right )-\frac {7 \arctan \left (\frac {\sqrt {x^3-1}}{\sqrt {3}}\right )}{8 \sqrt {3}}-\frac {\sqrt {x^3-1}}{3 x^3}-\frac {\sqrt {x^3-1}}{18 x^9}+\frac {5 \sqrt {x^3-1}}{36 x^6}\) |
-1/18*Sqrt[-1 + x^3]/x^9 + (5*Sqrt[-1 + x^3])/(36*x^6) - Sqrt[-1 + x^3]/(3 *x^3) + (13*ArcTan[Sqrt[-1 + x^3]])/24 - (7*ArcTan[Sqrt[-1 + x^3]/Sqrt[3]] )/(8*Sqrt[3])
3.10.3.3.1 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 = 5.52 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {12 x^{9}-17 x^{6}+7 x^{3}-2}{36 x^{9} \sqrt {x^{3}-1}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}-\frac {7 \arctan \left (\frac {\sqrt {x^{3}-1}\, \sqrt {3}}{3}\right ) \sqrt {3}}{24}\) | \(59\) |
pseudoelliptic | \(\frac {-21 \arctan \left (\frac {\sqrt {x^{3}-1}\, \sqrt {3}}{3}\right ) \sqrt {3}\, x^{9}+39 \arctan \left (\sqrt {x^{3}-1}\right ) x^{9}+\left (-24 x^{6}+10 x^{3}-4\right ) \sqrt {x^{3}-1}}{72 x^{9}}\) | \(61\) |
default | \(-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}-\frac {\sqrt {x^{3}-1}}{18 x^{9}}-\frac {7 \arctan \left (\frac {\sqrt {x^{3}-1}\, \sqrt {3}}{3}\right ) \sqrt {3}}{24}\) | \(66\) |
trager | \(-\frac {\left (12 x^{6}-5 x^{3}+2\right ) \sqrt {x^{3}-1}}{36 x^{9}}+\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{3}-1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{48}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{3}+6 \sqrt {x^{3}-1}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{3}+2}\right )}{48}\) | \(115\) |
elliptic | \(-\frac {\sqrt {x^{3}-1}}{18 x^{9}}+\frac {5 \sqrt {x^{3}-1}}{36 x^{6}}-\frac {\sqrt {x^{3}-1}}{3 x^{3}}+\frac {13 \arctan \left (\sqrt {x^{3}-1}\right )}{24}+\frac {7 \sqrt {2}\, \left (i \sqrt {3}+3\right ) \sqrt {-\frac {x -1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+2\right )}{\sum }\operatorname {EllipticPi}\left (\sqrt {-\frac {2 \left (x -1\right )}{i \sqrt {3}+3}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {i \sqrt {3}+3}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}\right )}{48 \sqrt {x^{3}-1}}+\frac {7 \sqrt {2}\, \left (i \sqrt {3}+3\right ) \sqrt {-\frac {x -1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+2\right )}{\sum }\operatorname {EllipticPi}\left (\sqrt {-\frac {2 \left (x -1\right )}{i \sqrt {3}+3}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {i \sqrt {3}+3}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{48 \sqrt {x^{3}-1}}+\frac {7 \sqrt {2}\, \left (i \sqrt {3}+3\right ) \sqrt {-\frac {x -1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}-2 x -1}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{3}+2\right )}{\sum }\operatorname {EllipticPi}\left (\sqrt {-\frac {2 \left (x -1\right )}{i \sqrt {3}+3}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {-\frac {i \sqrt {3}+3}{-3+i \sqrt {3}}}\right )\right )}{48 \sqrt {x^{3}-1}}\) | \(552\) |
-1/36*(12*x^9-17*x^6+7*x^3-2)/x^9/(x^3-1)^(1/2)+13/24*arctan((x^3-1)^(1/2) )-7/24*arctan(1/3*(x^3-1)^(1/2)*3^(1/2))*3^(1/2)
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=-\frac {21 \, \sqrt {3} x^{9} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{3} - 1}\right ) - 39 \, x^{9} \arctan \left (\sqrt {x^{3} - 1}\right ) + 2 \, {\left (12 \, x^{6} - 5 \, x^{3} + 2\right )} \sqrt {x^{3} - 1}}{72 \, x^{9}} \]
-1/72*(21*sqrt(3)*x^9*arctan(1/3*sqrt(3)*sqrt(x^3 - 1)) - 39*x^9*arctan(sq rt(x^3 - 1)) + 2*(12*x^6 - 5*x^3 + 2)*sqrt(x^3 - 1))/x^9
Time = 100.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=- \frac {7 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{3} - 1}}{3} \right )}}{24} + \frac {13 \operatorname {atan}{\left (\sqrt {x^{3} - 1} \right )}}{24} - \frac {7 \sqrt {x^{3} - 1}}{24 x^{3}} + \frac {\left (2 - x^{3}\right ) \sqrt {x^{3} - 1}}{24 x^{6}} + \frac {\left (x^{3} - 1\right )^{\frac {3}{2}}}{18 x^{9}} \]
-7*sqrt(3)*atan(sqrt(3)*sqrt(x**3 - 1)/3)/24 + 13*atan(sqrt(x**3 - 1))/24 - 7*sqrt(x**3 - 1)/(24*x**3) + (2 - x**3)*sqrt(x**3 - 1)/(24*x**6) + (x**3 - 1)**(3/2)/(18*x**9)
\[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} + 1\right )} \sqrt {x^{3} - 1}}{{\left (x^{3} + 2\right )} x^{10}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=-\frac {7}{24} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{3} - 1}\right ) - \frac {12 \, {\left (x^{3} - 1\right )}^{\frac {5}{2}} + 19 \, {\left (x^{3} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{3} - 1}}{36 \, x^{9}} + \frac {13}{24} \, \arctan \left (\sqrt {x^{3} - 1}\right ) \]
-7/24*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(x^3 - 1)) - 1/36*(12*(x^3 - 1)^(5/2) + 19*(x^3 - 1)^(3/2) + 9*sqrt(x^3 - 1))/x^9 + 13/24*arctan(sqrt(x^3 - 1))
Time = 7.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt {-1+x^3} \left (1-x^3+x^6\right )}{x^{10} \left (2+x^3\right )} \, dx=\frac {5\,\sqrt {x^3-1}}{36\,x^6}-\frac {\sqrt {x^3-1}}{3\,x^3}-\frac {\sqrt {x^3-1}}{18\,x^9}+\frac {\ln \left (\frac {\left (\sqrt {x^3-1}-\mathrm {i}\right )\,{\left (\sqrt {x^3-1}+1{}\mathrm {i}\right )}^3}{x^6}\right )\,13{}\mathrm {i}}{48}+\frac {\sqrt {3}\,\ln \left (\frac {6\,\sqrt {x^3-1}-\sqrt {3}\,4{}\mathrm {i}+\sqrt {3}\,x^3\,1{}\mathrm {i}}{x^3+2}\right )\,7{}\mathrm {i}}{48} \]