Integrand size = 28, antiderivative size = 26 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\sqrt {-1+x^4}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^4}}\right ) \] Output:
(x^4-1)^(1/2)/x+arctan(x/(x^4-1)^(1/2))
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\sqrt {-1+x^4}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^4}}\right ) \] Input:
Integrate[(Sqrt[-1 + x^4]*(1 + x^4))/(x^2*(-1 + x^2 + x^4)),x]
Output:
Sqrt[-1 + x^4]/x + ArcTan[x/Sqrt[-1 + x^4]]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.88 (sec) , antiderivative size = 251, normalized size of antiderivative = 9.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
\(\displaystyle \int \frac {\sqrt {x^4-1} \left (x^4+1\right )}{x^2 \left (x^4+x^2-1\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (2 x^2+1\right ) \sqrt {x^4-1}}{x^4+x^2-1}-\frac {\sqrt {x^4-1}}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1+\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{2 \sqrt {x^4-1}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{2 \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {x^4-1}}{x}\) |
Input:
Int[(Sqrt[-1 + x^4]*(1 + x^4))/(x^2*(-1 + x^2 + x^4)),x]
Output:
Sqrt[-1 + x^4]/x + ((1 - Sqrt[5])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticF[Ar cSin[x], -1])/(2*Sqrt[-1 + x^4]) + ((1 + Sqrt[5])*Sqrt[1 - x^2]*Sqrt[1 + x ^2]*EllipticF[ArcSin[x], -1])/(2*Sqrt[-1 + x^4]) - (Sqrt[2]*Sqrt[-1 + x^2] *Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/Sqrt[-1 + x^4] + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(1 - Sqrt[5])/2, ArcSin[ x], -1])/Sqrt[-1 + x^4] + (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(1 + Sqr t[5])/2, ArcSin[x], -1])/Sqrt[-1 + x^4]
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 = 1.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\sqrt {x^{4}-1}}{x}-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\) | \(27\) |
default | \(\frac {-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right ) x +\sqrt {x^{4}-1}}{x}\) | \(28\) |
pseudoelliptic | \(\frac {-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right ) x +\sqrt {x^{4}-1}}{x}\) | \(28\) |
elliptic | \(\frac {\left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(38\) |
trager | \(\frac {\sqrt {x^{4}-1}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}-1}\right )}{2}\) | \(72\) |
Input:
int((x^4-1)^(1/2)*(x^4+1)/x^2/(x^4+x^2-1),x,method=_RETURNVERBOSE)
Output:
(x^4-1)^(1/2)/x-arctan((x^4-1)^(1/2)/x)
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {x \arctan \left (\frac {2 \, \sqrt {x^{4} - 1} x}{x^{4} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{4} - 1}}{2 \, x} \] Input:
integrate((x^4-1)^(1/2)*(x^4+1)/x^2/(x^4+x^2-1),x, algorithm="fricas")
Output:
1/2*(x*arctan(2*sqrt(x^4 - 1)*x/(x^4 - x^2 - 1)) + 2*sqrt(x^4 - 1))/x
\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{x^{2} \left (x^{4} + x^{2} - 1\right )}\, dx \] Input:
integrate((x**4-1)**(1/2)*(x**4+1)/x**2/(x**4+x**2-1),x)
Output:
Integral(sqrt((x - 1)*(x + 1)*(x**2 + 1))*(x**4 + 1)/(x**2*(x**4 + x**2 - 1)), x)
\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}{{\left (x^{4} + x^{2} - 1\right )} x^{2}} \,d x } \] Input:
integrate((x^4-1)^(1/2)*(x^4+1)/x^2/(x^4+x^2-1),x, algorithm="maxima")
Output:
integrate((x^4 + 1)*sqrt(x^4 - 1)/((x^4 + x^2 - 1)*x^2), x)
\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}{{\left (x^{4} + x^{2} - 1\right )} x^{2}} \,d x } \] Input:
integrate((x^4-1)^(1/2)*(x^4+1)/x^2/(x^4+x^2-1),x, algorithm="giac")
Output:
integrate((x^4 + 1)*sqrt(x^4 - 1)/((x^4 + x^2 - 1)*x^2), x)
Timed out. \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {x^4-1}\,\left (x^4+1\right )}{x^2\,\left (x^4+x^2-1\right )} \,d x \] Input:
int(((x^4 - 1)^(1/2)*(x^4 + 1))/(x^2*(x^2 + x^4 - 1)),x)
Output:
int(((x^4 - 1)^(1/2)*(x^4 + 1))/(x^2*(x^2 + x^4 - 1)), x)
\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\sqrt {x^{4}-1}-\left (\int \frac {\sqrt {x^{4}-1}}{x^{8}+x^{6}-2 x^{4}-x^{2}+1}d x \right ) x -\left (\int \frac {\sqrt {x^{4}-1}\, x^{4}}{x^{8}+x^{6}-2 x^{4}-x^{2}+1}d x \right ) x}{x} \] Input:
int((x^4-1)^(1/2)*(x^4+1)/x^2/(x^4+x^2-1),x)
Output:
(sqrt(x**4 - 1) - int(sqrt(x**4 - 1)/(x**8 + x**6 - 2*x**4 - x**2 + 1),x)* x - int((sqrt(x**4 - 1)*x**4)/(x**8 + x**6 - 2*x**4 - x**2 + 1),x)*x)/x