Integrand size = 22, antiderivative size = 85 \[ \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx=-\frac {1}{3} \arctan \left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{6 \sqrt {2}}-\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{6 \sqrt {2}} \]
-1/3*arctan(x/(x^4+1)^(1/2))-1/12*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)- 1/3*arctanh(x/(x^4+1)^(1/2))-1/12*arctanh(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)
Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx=\frac {1}{12} \left (-4 \arctan \left (\frac {x}{\sqrt {1+x^4}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt {1+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )\right ) \]
(-4*ArcTan[x/Sqrt[1 + x^4]] - Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]] - 4*ArcTanh[x/Sqrt[1 + x^4]] - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]])/1 2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.76 (sec) , antiderivative size = 550, normalized size of antiderivative = 6.47, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2019, 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.
| \(\displaystyle \int \frac {x^{12}+1}{\sqrt {x^4+1} \left (x^{12}-1\right )} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {\sqrt {x^4+1} \left (x^8-x^4+1\right )}{x^{12}-1}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {\sqrt {x^4+1}}{12 (1-x)}-\frac {\sqrt {x^4+1}}{12 (1-i x)}-\frac {\sqrt {x^4+1}}{12 (1+i x)}-\frac {\sqrt {x^4+1}}{12 (x+1)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (1-\sqrt [6]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (\sqrt [6]{-1} x+1\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (\sqrt [3]{-1} x+1\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left ((-1)^{2/3} x+1\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left (1-(-1)^{5/6} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \sqrt {x^4+1}}{12 \left ((-1)^{5/6} x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \arctan \left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}-\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \sqrt {x^4+1}}-\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{6 \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (1-(-1)^{2/3}\right ) \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}}+\frac {\left (1-(-1)^{2/3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{12 \left (1+(-1)^{2/3}\right ) \sqrt {x^4+1}}+\frac {i \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4},2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {3} \sqrt {x^4+1}}+\frac {\left (\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan (x),\frac {1}{2}\right )}{12 \left (-\sqrt {3}+3 i\right ) \sqrt {x^4+1}}-\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan (x),\frac {1}{2}\right )}{24 \left (1+\sqrt [3]{-1}\right ) \sqrt {x^4+1}}-\frac {1}{3} \text {arctanh}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{6 \sqrt {2}}\) |
-1/3*ArcTan[x/Sqrt[1 + x^4]] - ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sqrt[2 ]) - ArcTanh[x/Sqrt[1 + x^4]]/3 - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(6*Sq rt[2]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2 ])/(3*(1 + (-1)^(1/3))*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2 )^2]*EllipticF[2*ArcTan[x], 1/2])/(3*(1 - (-1)^(2/3))*Sqrt[1 + x^4]) - ((1 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6*Sqrt[1 + x^4]) - ((1 + I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(6*Sqrt[1 + x^4]) + ((I/4)*(1 + x^2)* Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[1/4, 2*ArcTan[x], 1/2])/(Sqrt[3]*Sq rt[1 + x^4]) + ((1 - (-1)^(2/3))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Ell ipticPi[1/4, 2*ArcTan[x], 1/2])/(12*(1 + (-1)^(2/3))*Sqrt[1 + x^4]) - ((1 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[3/4, 2*ArcTa n[x], 1/2])/(24*(1 + (-1)^(1/3))*Sqrt[1 + x^4]) + ((I + Sqrt[3])*(1 + x^2) *Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[3/4, 2*ArcTan[x], 1/2])/(12*(3*I - Sqrt[3])*Sqrt[1 + x^4])
3.12.52.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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 = 7.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}+\frac {\ln \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-1\right )}{12}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{12}\right ) \sqrt {2}}{2}\) | \(99\) |
| default | \(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{12}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{24}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{24}-\frac {\arctan \left (\frac {\left (1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\arctan \left (\frac {\left (x -1\right )^{2}}{\sqrt {x^{4}+1}}\right )}{6}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {3}\, x^{2}+\sqrt {3}-2 x}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\operatorname {arctanh}\left (\frac {2 x +\sqrt {3}\, x^{2}+\sqrt {3}}{\sqrt {x^{4}+1}}\right )}{6}\) | \(150\) |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{12}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{24}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )}{24}-\frac {\arctan \left (\frac {\left (1+x \right )^{2}}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\arctan \left (\frac {\left (x -1\right )^{2}}{\sqrt {x^{4}+1}}\right )}{6}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {3}\, x^{2}+\sqrt {3}-2 x}{\sqrt {x^{4}+1}}\right )}{6}+\frac {\operatorname {arctanh}\left (\frac {2 x +\sqrt {3}\, x^{2}+\sqrt {3}}{\sqrt {x^{4}+1}}\right )}{6}\) | \(150\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -\sqrt {x^{4}+1}}{\left (x -1\right ) \left (1+x \right )}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )-4 \sqrt {x^{4}+1}\, x}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{12}+\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+1}\, x -x^{2}-1}{x^{4}-x^{2}+1}\right )}{6}\) | \(203\) |
1/2*(1/3*2^(1/2)*arctan((x^4+1)^(1/2)/x)+1/12*ln(1/2*2^(1/2)/x*(x^4+1)^(1/ 2)-1)-1/3*2^(1/2)*arctanh((x^4+1)^(1/2)/x)+1/6*arctan(1/2*2^(1/2)/x*(x^4+1 )^(1/2))-1/12*ln(1+1/2*2^(1/2)/x*(x^4+1)^(1/2)))*2^(1/2)
Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.41 \[ \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx=-\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) + \frac {1}{24} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 1} x}{x^{4} - x^{2} + 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 1} x + 1}{x^{4} - x^{2} + 1}\right ) \]
-1/12*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) + 1/24*sqrt(2)*log((x^4 - 2* sqrt(2)*sqrt(x^4 + 1)*x + 2*x^2 + 1)/(x^4 - 2*x^2 + 1)) - 1/6*arctan(2*sqr t(x^4 + 1)*x/(x^4 - x^2 + 1)) + 1/6*log((x^4 + x^2 - 2*sqrt(x^4 + 1)*x + 1 )/(x^4 - x^2 + 1))
\[ \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx=\int \frac {\sqrt {x^{4} + 1} \left (x^{8} - x^{4} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
Integral(sqrt(x**4 + 1)*(x**8 - x**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)*(x** 2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 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. \[ \int \frac {1+x^{12}}{\sqrt {1+x^4} \left (-1+x^{12}\right )} \, dx=\int \frac {x^{12}+1}{\sqrt {x^4+1}\,\left (x^{12}-1\right )} \,d x \]