Integrand size = 37, antiderivative size = 47 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+2 x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2+x^4}}\right ) \]
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+2 x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2+x^4}}\right ) \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.90 (sec) , antiderivative size = 1670, normalized size of antiderivative = 35.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, 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.
| \(\displaystyle \int \frac {\left (x^4+1\right ) \sqrt {x^4+2 x^2-1}}{\left (x^4-1\right ) \left (x^4+x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4+2 x^2-1} \left (-2 x^2-1\right )}{x^4+x^2-1}+\frac {\sqrt {x^4+2 x^2-1}}{x^2-1}+\frac {\sqrt {x^4+2 x^2-1}}{x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {3-2 \sqrt {2}} \left (1-\sqrt {5}\right ) \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {x^4+2 x^2-1}}+\frac {\sqrt {3-2 \sqrt {2}} \left (1+\sqrt {5}\right ) \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {x^4+2 x^2-1}}+\frac {\sqrt {2 \left (3-2 \sqrt {2}\right )} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}+\frac {\sqrt {2} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {\left (1+\sqrt {2}\right ) x^2-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} x}{\sqrt {\left (1+\sqrt {2}\right ) x^2-1}}\right ),\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {x^4+2 x^2-1}}-\frac {2 \sqrt {3-2 \sqrt {2}} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticPi}\left (1-\sqrt {2},\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}-\frac {2 \sqrt {3-2 \sqrt {2}} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticPi}\left (-1+\sqrt {2},\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}+\frac {\sqrt {3-2 \sqrt {2}} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticPi}\left (\frac {2 \left (1-\sqrt {2}\right )}{1-\sqrt {5}},\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}+\frac {\sqrt {3-2 \sqrt {2}} \sqrt {x^2+\sqrt {2}+1} \sqrt {1-\left (1+\sqrt {2}\right ) x^2} \operatorname {EllipticPi}\left (\frac {2 \left (1-\sqrt {2}\right )}{1+\sqrt {5}},\arcsin \left (\sqrt {1+\sqrt {2}} x\right ),-3+2 \sqrt {2}\right )}{\sqrt {x^4+2 x^2-1}}\) |
(Sqrt[2]*Sqrt[1 + Sqrt[2] + x^2]*Sqrt[1 - (1 + Sqrt[2])*x^2]*EllipticF[Arc Sin[Sqrt[1 + Sqrt[2]]*x], -3 + 2*Sqrt[2]])/Sqrt[-1 + 2*x^2 + x^4] + (Sqrt[ 2*(3 - 2*Sqrt[2])]*Sqrt[1 + Sqrt[2] + x^2]*Sqrt[1 - (1 + Sqrt[2])*x^2]*Ell ipticF[ArcSin[Sqrt[1 + Sqrt[2]]*x], -3 + 2*Sqrt[2]])/Sqrt[-1 + 2*x^2 + x^4 ] + (Sqrt[3 - 2*Sqrt[2]]*(1 + Sqrt[5])*Sqrt[1 + Sqrt[2] + x^2]*Sqrt[1 - (1 + Sqrt[2])*x^2]*EllipticF[ArcSin[Sqrt[1 + Sqrt[2]]*x], -3 + 2*Sqrt[2]])/( (1 - 2*Sqrt[2] - Sqrt[5])*Sqrt[-1 + 2*x^2 + x^4]) + (Sqrt[3 - 2*Sqrt[2]]*( 1 - Sqrt[5])*Sqrt[1 + Sqrt[2] + x^2]*Sqrt[1 - (1 + Sqrt[2])*x^2]*EllipticF [ArcSin[Sqrt[1 + Sqrt[2]]*x], -3 + 2*Sqrt[2]])/((1 - 2*Sqrt[2] + Sqrt[5])* Sqrt[-1 + 2*x^2 + x^4]) - (2^(1/4)*Sqrt[(1 - (1 - Sqrt[2])*x^2)/(1 - (1 + Sqrt[2])*x^2)]*Sqrt[-1 + (1 + Sqrt[2])*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/S qrt[-1 + (1 + Sqrt[2])*x^2]], (2 + Sqrt[2])/4])/((2 - Sqrt[2])*Sqrt[(1 - ( 1 + Sqrt[2])*x^2)^(-1)]*Sqrt[-1 + 2*x^2 + x^4]) + ((1 + Sqrt[2])*Sqrt[(1 - (1 - Sqrt[2])*x^2)/(1 - (1 + Sqrt[2])*x^2)]*Sqrt[-1 + (1 + Sqrt[2])*x^2]* EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[-1 + (1 + Sqrt[2])*x^2]], (2 + Sqrt[2])/ 4])/(2^(1/4)*Sqrt[(1 - (1 + Sqrt[2])*x^2)^(-1)]*Sqrt[-1 + 2*x^2 + x^4]) - ((1 + 2*Sqrt[2] - Sqrt[5])*Sqrt[(1 - (1 - Sqrt[2])*x^2)/(1 - (1 + Sqrt[2]) *x^2)]*Sqrt[-1 + (1 + Sqrt[2])*x^2]*EllipticF[ArcSin[(2^(3/4)*x)/Sqrt[-1 + (1 + Sqrt[2])*x^2]], (2 + Sqrt[2])/4])/(2*2^(3/4)*Sqrt[(1 - (1 + Sqrt[2]) *x^2)^(-1)]*Sqrt[-1 + 2*x^2 + x^4]) - ((1 + Sqrt[5])*Sqrt[(1 - (1 - Sqr...
3.6.97.3.1 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 = 3.92 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.62
| method | result | size |
| elliptic | \(\frac {\left (\ln \left (\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}-1\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+2 x^{2}-1}}{x}\right )-\ln \left (1+\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(76\) |
| default | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\left (1+i\right ) x^{2}+\left (2-2 i\right ) x -1+i\right )}{2 \sqrt {x^{4}+2 x^{2}-1}}\right )}{2}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\left (1+i\right ) x^{2}+\left (-2+2 i\right ) x -1+i\right ) \sqrt {2}}{2 \sqrt {x^{4}+2 x^{2}-1}}\right )}{2}+\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+2 x^{2}-1}}{x}\right )\) | \(97\) |
| pseudoelliptic | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\left (1+i\right ) x^{2}+\left (2-2 i\right ) x -1+i\right )}{2 \sqrt {x^{4}+2 x^{2}-1}}\right )}{2}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\left (1+i\right ) x^{2}+\left (-2+2 i\right ) x -1+i\right ) \sqrt {2}}{2 \sqrt {x^{4}+2 x^{2}-1}}\right )}{2}+\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+2 x^{2}-1}}{x}\right )\) | \(97\) |
| trager | \(-\frac {\ln \left (-\frac {-x^{4}+2 x \sqrt {x^{4}+2 x^{2}-1}-3 x^{2}+1}{x^{4}+x^{2}-1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 x \sqrt {x^{4}+2 x^{2}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(116\) |
1/2*(ln(1/2*(x^4+2*x^2-1)^(1/2)*2^(1/2)/x-1)+2^(1/2)*arctanh((x^4+2*x^2-1) ^(1/2)/x)-ln(1+1/2*(x^4+2*x^2-1)^(1/2)*2^(1/2)/x))*2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (39) = 78\).
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.32 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{8} + 16 \, x^{6} + 30 \, x^{4} - 4 \, \sqrt {2} {\left (x^{5} + 4 \, x^{3} - x\right )} \sqrt {x^{4} + 2 \, x^{2} - 1} - 16 \, x^{2} + 1}{x^{8} - 2 \, x^{4} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + 3 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{2} - 1} x - 1}{x^{4} + x^{2} - 1}\right ) \]
1/4*sqrt(2)*log(-(x^8 + 16*x^6 + 30*x^4 - 4*sqrt(2)*(x^5 + 4*x^3 - x)*sqrt (x^4 + 2*x^2 - 1) - 16*x^2 + 1)/(x^8 - 2*x^4 + 1)) + 1/2*log((x^4 + 3*x^2 + 2*sqrt(x^4 + 2*x^2 - 1)*x - 1)/(x^4 + x^2 - 1))
\[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^{4} + 1\right ) \sqrt {x^{4} + 2 x^{2} - 1}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + x^{2} - 1\right )}\, dx \]
Integral((x**4 + 1)*sqrt(x**4 + 2*x**2 - 1)/((x - 1)*(x + 1)*(x**2 + 1)*(x **4 + x**2 - 1)), x)
\[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}} \,d x } \]
\[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\int { \frac {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx=\int \frac {\left (x^4+1\right )\,\sqrt {x^4+2\,x^2-1}}{\left (x^4-1\right )\,\left (x^4+x^2-1\right )} \,d x \]