Integrand size = 38, antiderivative size = 84 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+4 x^3+x^6\right ) \sqrt {1+x^{12}}}{6 x^6}-\frac {4 \text {arctanh}\left (\frac {\sqrt {3} x^3}{-1-x^3+x^6+\sqrt {1+x^{12}}}\right )}{\sqrt {3}}-3 \log (x)+\log \left (-1+x^6+\sqrt {1+x^{12}}\right ) \]
1/6*(x^6+4*x^3-1)*(x^12+1)^(1/2)/x^6-4/3*arctanh(3^(1/2)*x^3/(-1-x^3+x^6+( x^12+1)^(1/2)))*3^(1/2)-3*ln(x)+ln(-1+x^6+(x^12+1)^(1/2))
\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.97 (sec) , antiderivative size = 564, normalized size of antiderivative = 6.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {7279, 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.
| \(\displaystyle \int \frac {\left (x^6+1\right ) \left (x^6+x^3-1\right ) \sqrt {x^{12}+1}}{x^7 \left (x^6-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {3 \sqrt {x^{12}+1}}{x}+\frac {\sqrt {x^{12}+1}}{x^7}-\frac {2 \sqrt {x^{12}+1}}{x^4}-\frac {2 \left (x^3-3\right ) \sqrt {x^{12}+1} x^2}{x^6-x^3-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (1+\sqrt {5}\right ) \text {arcsinh}\left (x^6\right )+\frac {1}{6} \left (1-\sqrt {5}\right ) \text {arcsinh}\left (x^6\right )+\frac {\text {arcsinh}\left (x^6\right )}{6}+\frac {2 \left (3+\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^3\right ),\frac {1}{2}\right )}{3 \left (5+\sqrt {5}\right ) \sqrt {x^{12}+1}}+\frac {2 \left (3-\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^3\right ),\frac {1}{2}\right )}{3 \left (5-\sqrt {5}\right ) \sqrt {x^{12}+1}}-\frac {2 \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^3\right ),\frac {1}{2}\right )}{3 \sqrt {x^{12}+1}}+\frac {\left (3+\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \operatorname {EllipticPi}\left (\frac {5}{4},2 \arctan \left (x^3\right ),\frac {1}{2}\right )}{2 \left (5+3 \sqrt {5}\right ) \sqrt {x^{12}+1}}+\frac {\left (3-\sqrt {5}\right ) \left (x^6+1\right ) \sqrt {\frac {x^{12}+1}{\left (x^6+1\right )^2}} \operatorname {EllipticPi}\left (\frac {5}{4},2 \arctan \left (x^3\right ),\frac {1}{2}\right )}{2 \left (5-3 \sqrt {5}\right ) \sqrt {x^{12}+1}}-\frac {1}{2} \text {arctanh}\left (\sqrt {x^{12}+1}\right )+\frac {\text {arctanh}\left (\frac {\left (3-\sqrt {5}\right ) x^6+2}{\sqrt {6 \left (3-\sqrt {5}\right )} \sqrt {x^{12}+1}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {\left (3+\sqrt {5}\right ) x^6+2}{\sqrt {6 \left (3+\sqrt {5}\right )} \sqrt {x^{12}+1}}\right )}{\sqrt {3}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} x^3}{\sqrt {x^{12}+1}}\right )}{\sqrt {3}}-\frac {1}{6} \left (1+\sqrt {5}\right ) \sqrt {x^{12}+1}-\frac {1}{6} \left (1-\sqrt {5}\right ) \sqrt {x^{12}+1}+\frac {\sqrt {x^{12}+1}}{2}-\frac {\sqrt {x^{12}+1}}{6 x^6}+\frac {2 \sqrt {x^{12}+1}}{3 x^3}\) |
Sqrt[1 + x^12]/2 - ((1 - Sqrt[5])*Sqrt[1 + x^12])/6 - ((1 + Sqrt[5])*Sqrt[ 1 + x^12])/6 - Sqrt[1 + x^12]/(6*x^6) + (2*Sqrt[1 + x^12])/(3*x^3) + ArcSi nh[x^6]/6 + ((1 - Sqrt[5])*ArcSinh[x^6])/6 + ((1 + Sqrt[5])*ArcSinh[x^6])/ 6 - (2*ArcTanh[(Sqrt[3]*x^3)/Sqrt[1 + x^12]])/Sqrt[3] + ArcTanh[(2 + (3 - Sqrt[5])*x^6)/(Sqrt[6*(3 - Sqrt[5])]*Sqrt[1 + x^12])]/Sqrt[3] - ArcTanh[(2 + (3 + Sqrt[5])*x^6)/(Sqrt[6*(3 + Sqrt[5])]*Sqrt[1 + x^12])]/Sqrt[3] - Ar cTanh[Sqrt[1 + x^12]]/2 - (2*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*Ellipt icF[2*ArcTan[x^3], 1/2])/(3*Sqrt[1 + x^12]) + (2*(3 - Sqrt[5])*(1 + x^6)*S qrt[(1 + x^12)/(1 + x^6)^2]*EllipticF[2*ArcTan[x^3], 1/2])/(3*(5 - Sqrt[5] )*Sqrt[1 + x^12]) + (2*(3 + Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2 ]*EllipticF[2*ArcTan[x^3], 1/2])/(3*(5 + Sqrt[5])*Sqrt[1 + x^12]) + ((3 - Sqrt[5])*(1 + x^6)*Sqrt[(1 + x^12)/(1 + x^6)^2]*EllipticPi[5/4, 2*ArcTan[x ^3], 1/2])/(2*(5 - 3*Sqrt[5])*Sqrt[1 + x^12]) + ((3 + Sqrt[5])*(1 + x^6)*S qrt[(1 + x^12)/(1 + x^6)^2]*EllipticPi[5/4, 2*ArcTan[x^3], 1/2])/(2*(5 + 3 *Sqrt[5])*Sqrt[1 + x^12])
3.12.29.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.64 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-4 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (x^{6}+2 x^{3}-1\right ) \sqrt {3}}{3 x \sqrt {\frac {x^{12}+1}{x^{2}}}}\right ) x^{5}+6 \,\operatorname {arcsinh}\left (\frac {\left (x^{6}-1\right ) \sqrt {2}}{2 x^{3}}\right ) x^{5}+\sqrt {\frac {x^{12}+1}{x^{2}}}\, \left (x^{6}+4 x^{3}-1\right )}{6 x^{5}}\) | \(86\) |
| trager | \(\frac {\left (x^{6}+4 x^{3}-1\right ) \sqrt {x^{12}+1}}{6 x^{6}}+\ln \left (\frac {-1+x^{6}+\sqrt {x^{12}+1}}{x^{3}}\right )-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+3 \sqrt {x^{12}+1}}{x^{6}-x^{3}-1}\right )}{3}\) | \(102\) |
| risch | \(\frac {4 x^{15}-x^{12}+4 x^{3}-1}{6 x^{6} \sqrt {x^{12}+1}}+\frac {\sqrt {x^{12}+1}}{6}+\ln \left (\frac {-1+x^{6}+\sqrt {x^{12}+1}}{x^{3}}\right )+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )-3 \sqrt {x^{12}+1}}{x^{6}-x^{3}-1}\right )}{3}\) | \(119\) |
1/6*(-4*3^(1/2)*arctanh(1/3*(x^6+2*x^3-1)/x*3^(1/2)/(1/x^2*(x^12+1))^(1/2) )*x^5+6*arcsinh(1/2*(x^6-1)/x^3*2^(1/2))*x^5+(1/x^2*(x^12+1))^(1/2)*(x^6+4 *x^3-1))/x^5
Time = 0.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.43 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\frac {2 \, \sqrt {3} x^{6} \log \left (\frac {2 \, x^{12} + 2 \, x^{9} + x^{6} - 2 \, x^{3} - \sqrt {3} \sqrt {x^{12} + 1} {\left (x^{6} + 2 \, x^{3} - 1\right )} + 2}{x^{12} - 2 \, x^{9} - x^{6} + 2 \, x^{3} + 1}\right ) + 6 \, x^{6} \log \left (\frac {x^{6} + \sqrt {x^{12} + 1} - 1}{x^{3}}\right ) + \sqrt {x^{12} + 1} {\left (x^{6} + 4 \, x^{3} - 1\right )}}{6 \, x^{6}} \]
1/6*(2*sqrt(3)*x^6*log((2*x^12 + 2*x^9 + x^6 - 2*x^3 - sqrt(3)*sqrt(x^12 + 1)*(x^6 + 2*x^3 - 1) + 2)/(x^12 - 2*x^9 - x^6 + 2*x^3 + 1)) + 6*x^6*log(( x^6 + sqrt(x^12 + 1) - 1)/x^3) + sqrt(x^12 + 1)*(x^6 + 4*x^3 - 1))/x^6
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {\sqrt {x^{12} + 1} {\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{7}} \,d x } \]
\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {\sqrt {x^{12} + 1} {\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (x^{6} - x^{3} - 1\right )} x^{7}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right ) \sqrt {1+x^{12}}}{x^7 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {\left (x^6+1\right )\,\sqrt {x^{12}+1}\,\left (x^6+x^3-1\right )}{x^7\,\left (-x^6+x^3+1\right )} \,d x \]