Integrand size = 29, antiderivative size = 26 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\frac {\arctan \left (\frac {x \sqrt {-3+x^4}}{\sqrt {2}}\right )}{3 \sqrt {2}} \] Output:
1/6*arctan(1/2*x*(x^4-3)^(1/2)*2^(1/2))*2^(1/2)
Time = 9.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\frac {\arctan \left (\frac {x \sqrt {-3+x^4}}{\sqrt {2}}\right )}{3 \sqrt {2}} \] Input:
Integrate[(1 + x^2)/((-1 + x^2)*(2 + x^2)*Sqrt[-3 + x^4]),x]
Output:
ArcTan[(x*Sqrt[-3 + x^4])/Sqrt[2]]/(3*Sqrt[2])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.86 (sec) , antiderivative size = 513, normalized size of antiderivative = 19.73, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int \frac {x^2+1}{\left (x^2-1\right ) \left (x^2+2\right ) \sqrt {x^4-3}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2}{3 \left (x^2-1\right ) \sqrt {x^4-3}}+\frac {1}{3 \left (x^2+2\right ) \sqrt {x^4-3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {\sqrt {3}-x^2} \sqrt {x^2+\sqrt {3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{3 \sqrt [4]{3} \left (2+\sqrt {3}\right ) \sqrt {x^4-3}}+\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {x^2+\sqrt {3}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{3 \sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt {x^4-3}}+\frac {\sqrt {\frac {x^2+\sqrt {3}}{\sqrt {3}-x^2}} \sqrt {\sqrt {3} x^2-3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {\sqrt {3} x^2-3}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {x^4-3}}-\frac {\sqrt {2} \sqrt {\frac {x^2+\sqrt {3}}{\sqrt {3}-x^2}} \sqrt {\sqrt {3} x^2-3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {\sqrt {3} x^2-3}}\right ),\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {x^4-3}}+\frac {\sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \operatorname {EllipticPi}\left (-\frac {\sqrt {3}}{2},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{6 \sqrt {3} \sqrt {x^4-3}}-\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \operatorname {EllipticPi}\left (\sqrt {3},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{3 \sqrt {3} \sqrt {x^4-3}}\) |
Input:
Int[(1 + x^2)/((-1 + x^2)*(2 + x^2)*Sqrt[-3 + x^4]),x]
Output:
(2*Sqrt[Sqrt[3] - x^2]*Sqrt[Sqrt[3] + x^2]*EllipticF[ArcSin[x/3^(1/4)], -1 ])/(3*3^(1/4)*(1 - Sqrt[3])*Sqrt[-3 + x^4]) - (Sqrt[Sqrt[3] - x^2]*Sqrt[Sq rt[3] + x^2]*EllipticF[ArcSin[x/3^(1/4)], -1])/(3*3^(1/4)*(2 + Sqrt[3])*Sq rt[-3 + x^4]) - (Sqrt[2]*Sqrt[(Sqrt[3] + x^2)/(Sqrt[3] - x^2)]*Sqrt[-3 + S qrt[3]*x^2]*EllipticF[ArcSin[(Sqrt[2]*3^(1/4)*x)/Sqrt[-3 + Sqrt[3]*x^2]], 1/2])/(3*3^(3/4)*(1 - Sqrt[3])*Sqrt[(3 - Sqrt[3]*x^2)^(-1)]*Sqrt[-3 + x^4] ) + (Sqrt[(Sqrt[3] + x^2)/(Sqrt[3] - x^2)]*Sqrt[-3 + Sqrt[3]*x^2]*Elliptic F[ArcSin[(Sqrt[2]*3^(1/4)*x)/Sqrt[-3 + Sqrt[3]*x^2]], 1/2])/(3*Sqrt[2]*3^( 3/4)*(2 + Sqrt[3])*Sqrt[(3 - Sqrt[3]*x^2)^(-1)]*Sqrt[-3 + x^4]) + (Sqrt[Sq rt[3] - x^2]*Sqrt[3 + Sqrt[3]*x^2]*EllipticPi[-1/2*Sqrt[3], ArcSin[x/3^(1/ 4)], -1])/(6*Sqrt[3]*Sqrt[-3 + x^4]) - (2*Sqrt[Sqrt[3] - x^2]*Sqrt[3 + Sqr t[3]*x^2]*EllipticPi[Sqrt[3], ArcSin[x/3^(1/4)], -1])/(3*Sqrt[3]*Sqrt[-3 + x^4])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.74 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-3}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+2\right ) \left (1+x \right )^{2} \left (-1+x \right )^{2}}\right )}{12}\) | \(68\) |
elliptic | \(-\frac {2 \sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , -\sqrt {3}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{3 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}-\frac {\sqrt {\sqrt {3}\, x^{2}+3}\, \sqrt {-\sqrt {3}\, x^{2}+3}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , \frac {\sqrt {3}}{2}, i\right )}{18 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}\) | \(125\) |
default | \(-\frac {2 \sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , -\sqrt {3}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{3 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}+\frac {\sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , \frac {\sqrt {3}}{2}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{6 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}\) | \(136\) |
Input:
int((x^2+1)/(x^2-1)/(x^2+2)/(x^4-3)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/12*RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^6-3*RootOf(_Z^2+2)*x^2+4*(x^4-3) ^(1/2)*x-2*RootOf(_Z^2+2))/(x^2+2)/(1+x)^2/(-1+x)^2)
Exception generated. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((x^2+1)/(x^2-1)/(x^2+2)/(x^4-3)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\int \frac {x^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 2\right ) \sqrt {x^{4} - 3}}\, dx \] Input:
integrate((x**2+1)/(x**2-1)/(x**2+2)/(x**4-3)**(1/2),x)
Output:
Integral((x**2 + 1)/((x - 1)*(x + 1)*(x**2 + 2)*sqrt(x**4 - 3)), x)
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{4} - 3} {\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate((x^2+1)/(x^2-1)/(x^2+2)/(x^4-3)^(1/2),x, algorithm="maxima")
Output:
integrate((x^2 + 1)/(sqrt(x^4 - 3)*(x^2 + 2)*(x^2 - 1)), x)
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{4} - 3} {\left (x^{2} + 2\right )} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate((x^2+1)/(x^2-1)/(x^2+2)/(x^4-3)^(1/2),x, algorithm="giac")
Output:
integrate((x^2 + 1)/(sqrt(x^4 - 3)*(x^2 + 2)*(x^2 - 1)), x)
Timed out. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\int \frac {x^2+1}{\left (x^2-1\right )\,\left (x^2+2\right )\,\sqrt {x^4-3}} \,d x \] Input:
int((x^2 + 1)/((x^2 - 1)*(x^2 + 2)*(x^4 - 3)^(1/2)),x)
Output:
int((x^2 + 1)/((x^2 - 1)*(x^2 + 2)*(x^4 - 3)^(1/2)), x)
\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\int \frac {\sqrt {x^{4}-3}}{x^{8}+x^{6}-5 x^{4}-3 x^{2}+6}d x +\int \frac {\sqrt {x^{4}-3}\, x^{2}}{x^{8}+x^{6}-5 x^{4}-3 x^{2}+6}d x \] Input:
int((x^2+1)/(x^2-1)/(x^2+2)/(x^4-3)^(1/2),x)
Output:
int(sqrt(x**4 - 3)/(x**8 + x**6 - 5*x**4 - 3*x**2 + 6),x) + int((sqrt(x**4 - 3)*x**2)/(x**8 + x**6 - 5*x**4 - 3*x**2 + 6),x)