Integrand size = 22, antiderivative size = 67 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}} \]
-1/3*x/(x^4+1)^(1/2)-1/9*arctan(3^(1/4)*x/(x^4+1)^(1/2))*3^(3/4)-1/9*arcta nh(3^(1/4)*x/(x^4+1)^(1/2))*3^(3/4)
Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {x}{3 \sqrt {1+x^4}}-\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {1+x^4}}\right )}{3 \sqrt [4]{3}} \]
-1/3*x/Sqrt[1 + x^4] - ArcTan[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4)) - Arc Tanh[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.34 (sec) , antiderivative size = 663, normalized size of antiderivative = 9.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {x^{12}-1}{\sqrt {x^4+1} \left (x^{12}+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+1}}-\frac {2}{\sqrt {x^4+1} \left (x^{12}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [4]{3} x}{\sqrt {x^4+1}}\right )}{3 \sqrt [4]{3}}+\frac {\left (\sqrt {3}+(-2+i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \left (1-i \sqrt {3}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (\sqrt {3}+(-2-i)\right ) \sqrt {x^4+1}}-\frac {\sqrt [6]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \left (1+\sqrt [6]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \left (1-\sqrt [6]{-1}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \sqrt {x^4+1}}+\frac {i \left (2+\sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt {3}\right ),2 \arctan (x),\frac {1}{2}\right )}{12 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{5/6}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt {3}\right ),2 \arctan (x),\frac {1}{2}\right )}{12 \left (1+(-1)^{5/6}\right ) \sqrt {x^4+1}}-\frac {\left ((-2-3 i)+(1+2 i) \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\sqrt {3}\right ),2 \arctan (x),\frac {1}{2}\right )}{12 \left (\sqrt {3}+i\right ) \sqrt {x^4+1}}-\frac {\left (1-\sqrt [6]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\sqrt {3}\right ),2 \arctan (x),\frac {1}{2}\right )}{12 \left (1+\sqrt [6]{-1}\right ) \sqrt {x^4+1}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt {x^4+1}}\right )}{3 \sqrt [4]{3}}+\frac {i x \left (-x^2+i\right )}{6 \sqrt {x^4+1}}+\frac {i x \left (x^2+i\right )}{6 \sqrt {x^4+1}}\) |
((I/6)*x*(I - x^2))/Sqrt[1 + x^4] + ((I/6)*x*(I + x^2))/Sqrt[1 + x^4] - Ar cTan[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4)) - ArcTanh[(3^(1/4)*x)/Sqrt[1 + x^4]]/(3*3^(1/4)) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*Ar cTan[x], 1/2])/(3*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]* EllipticF[2*ArcTan[x], 1/2])/(6*(1 - (-1)^(1/6))*Sqrt[1 + x^4]) - ((-1)^(1 /6)*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6* (1 + (-1)^(1/6))*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*E llipticF[2*ArcTan[x], 1/2])/(3*((-2 - I) + Sqrt[3])*Sqrt[1 + x^4]) + (((-2 + I) + Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[ x], 1/2])/(6*(1 - I*Sqrt[3])*Sqrt[1 + x^4]) + ((1 - (-1)^(5/6))*(1 + x^2)* Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(2 - Sqrt[3])/4, 2*ArcTan[x], 1/2]) /(12*(1 + (-1)^(5/6))*Sqrt[1 + x^4]) + ((I/12)*(2 + Sqrt[3])*(1 + x^2)*Sqr t[(1 + x^4)/(1 + x^2)^2]*EllipticPi[(2 - Sqrt[3])/4, 2*ArcTan[x], 1/2])/Sq rt[1 + x^4] - ((1 - (-1)^(1/6))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Elli pticPi[(2 + Sqrt[3])/4, 2*ArcTan[x], 1/2])/(12*(1 + (-1)^(1/6))*Sqrt[1 + x ^4]) - (((-2 - 3*I) + (1 + 2*I)*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2 )^2]*EllipticPi[(2 + Sqrt[3])/4, 2*ArcTan[x], 1/2])/(12*(I + Sqrt[3])*Sqrt [1 + x^4])
3.9.91.3.1 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 = 21.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(-\frac {x}{3 \sqrt {x^{4}+1}}-\frac {3^{\frac {3}{4}} \left (-2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}+1}}{3 x}\right )+\ln \left (\frac {-3^{\frac {1}{4}} x -\sqrt {x^{4}+1}}{3^{\frac {1}{4}} x -\sqrt {x^{4}+1}}\right )\right )}{18}\) | \(71\) |
| elliptic | \(\frac {\left (-\frac {\sqrt {2}\, x}{3 \sqrt {x^{4}+1}}+\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (2 \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}+1}}{3 x}\right )-\ln \left (\frac {\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}+\frac {3^{\frac {1}{4}} \sqrt {2}}{2}}{\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\frac {3^{\frac {1}{4}} \sqrt {2}}{2}}\right )\right )}{18}\right ) \sqrt {2}}{2}\) | \(101\) |
| default | \(\frac {-\left (x^{4}+1\right ) \left (\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )+\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )\right ) 3^{\frac {3}{4}}-6 \sqrt {x^{4}+1}\, x}{18 \left (x^{2}+x \sqrt {2}+1\right ) \left (x^{2}-x \sqrt {2}+1\right )}\) | \(178\) |
| pseudoelliptic | \(\frac {-\left (x^{4}+1\right ) \left (\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (1+\sqrt {3}\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )-\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}-4 x \right )}{6 \sqrt {x^{4}+1}}\right )+\arctan \left (\frac {3^{\frac {3}{4}} \left (\left (x^{2}+1\right ) \left (\sqrt {3}-1\right ) \sqrt {2}+4 x \right )}{6 \sqrt {x^{4}+1}}\right )\right ) 3^{\frac {3}{4}}-6 \sqrt {x^{4}+1}\, x}{18 \left (x^{2}+x \sqrt {2}+1\right ) \left (x^{2}-x \sqrt {2}+1\right )}\) | \(178\) |
| trager | \(-\frac {x}{3 \sqrt {x^{4}+1}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{3} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right ) x^{4}+18 \sqrt {x^{4}+1}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}-3 x^{4}-3}\right )}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right ) x^{4}+18 \sqrt {x^{4}+1}\, x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2}\right )}{3 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-27\right )^{2} x^{2}+3}\right )}{18}\) | \(195\) |
-1/3*x/(x^4+1)^(1/2)-1/18*3^(3/4)*(-2*arctan(1/3*3^(3/4)/x*(x^4+1)^(1/2))+ ln((-3^(1/4)*x-(x^4+1)^(1/2))/(3^(1/4)*x-(x^4+1)^(1/2))))
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 326, normalized size of antiderivative = 4.87 \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=-\frac {3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (-\frac {3^{\frac {3}{4}} {\left (x^{8} + 5 \, x^{4} + 1\right )} + 6 \, {\left (x^{5} + \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} + 6 \cdot 3^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (x^{8} + 5 \, x^{4} + 1\right )} - 6 \, {\left (x^{5} + \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} + 6 \cdot 3^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (i \, x^{8} + 5 i \, x^{4} + i\right )} - 6 \, {\left (x^{5} - \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} - 6 \cdot 3^{\frac {1}{4}} {\left (i \, x^{6} + i \, x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) - 3^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (\frac {3^{\frac {3}{4}} {\left (-i \, x^{8} - 5 i \, x^{4} - i\right )} - 6 \, {\left (x^{5} - \sqrt {3} x^{3} + x\right )} \sqrt {x^{4} + 1} - 6 \cdot 3^{\frac {1}{4}} {\left (-i \, x^{6} - i \, x^{2}\right )}}{x^{8} - x^{4} + 1}\right ) + 12 \, \sqrt {x^{4} + 1} x}{36 \, {\left (x^{4} + 1\right )}} \]
-1/36*(3^(3/4)*(x^4 + 1)*log(-(3^(3/4)*(x^8 + 5*x^4 + 1) + 6*(x^5 + sqrt(3 )*x^3 + x)*sqrt(x^4 + 1) + 6*3^(1/4)*(x^6 + x^2))/(x^8 - x^4 + 1)) - 3^(3/ 4)*(x^4 + 1)*log((3^(3/4)*(x^8 + 5*x^4 + 1) - 6*(x^5 + sqrt(3)*x^3 + x)*sq rt(x^4 + 1) + 6*3^(1/4)*(x^6 + x^2))/(x^8 - x^4 + 1)) - 3^(3/4)*(I*x^4 + I )*log((3^(3/4)*(I*x^8 + 5*I*x^4 + I) - 6*(x^5 - sqrt(3)*x^3 + x)*sqrt(x^4 + 1) - 6*3^(1/4)*(I*x^6 + I*x^2))/(x^8 - x^4 + 1)) - 3^(3/4)*(-I*x^4 - I)* log((3^(3/4)*(-I*x^8 - 5*I*x^4 - I) - 6*(x^5 - sqrt(3)*x^3 + x)*sqrt(x^4 + 1) - 6*3^(1/4)*(-I*x^6 - I*x^2))/(x^8 - x^4 + 1)) + 12*sqrt(x^4 + 1)*x)/( x^4 + 1)
\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\left (x^{4} + 1\right )^{\frac {3}{2}} \left (x^{8} - x^{4} + 1\right )}\, dx \]
Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1)/((x**4 + 1)**(3/2)*(x**8 - x**4 + 1)), x)
\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int { \frac {x^{12} - 1}{{\left (x^{12} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
\[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int { \frac {x^{12} - 1}{{\left (x^{12} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Timed out. \[ \int \frac {-1+x^{12}}{\sqrt {1+x^4} \left (1+x^{12}\right )} \, dx=\int \frac {x^{12}-1}{\sqrt {x^4+1}\,\left (x^{12}+1\right )} \,d x \]