Integrand size = 30, antiderivative size = 180 \[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {2 \sqrt {1+x^2+x^4}}{3 x^3}+\frac {\sqrt {1+x^2+x^4}}{3 x}-\frac {x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}} \] Output:
-2/3*(x^4+x^2+1)^(1/2)/x^3+1/3*(x^4+x^2+1)^(1/2)/x-x*(x^4+x^2+1)^(1/2)/(3* x^2+3)-1/2*arctan(x/(x^4+x^2+1)^(1/2))+1/3*(x^2+1)*((x^4+x^2+1)/(x^2+1)^2) ^(1/2)*EllipticE(sin(2*arctan(x)),1/2)/(x^4+x^2+1)^(1/2)-3/4*(x^2+1)*((x^4 +x^2+1)/(x^2+1)^2)^(1/2)*InverseJacobiAM(2*arctan(x),1/2)/(x^4+x^2+1)^(1/2 )
Result contains complex when optimal does not.
Time = 10.44 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.21 \[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {-2-x^2-x^4+x^6-\sqrt [3]{-1} x^3 \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-(-1)^{5/6} x^3 \sqrt {3+3 \sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-3 (-1)^{2/3} x^3 \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{3 x^3 \sqrt {1+x^2+x^4}} \] Input:
Integrate[(2 + 3*x^2)/(x^4*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
(-2 - x^2 - x^4 + x^6 - (-1)^(1/3)*x^3*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - ( -1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - (-1)^(5/6) *x^3*Sqrt[3 + 3*(-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticF[I*ArcSi nh[(-1)^(5/6)*x], (-1)^(2/3)] - 3*(-1)^(2/3)*x^3*Sqrt[1 + (-1)^(1/3)*x^2]* Sqrt[1 - (-1)^(2/3)*x^2]*EllipticPi[(-1)^(1/3), I*ArcSinh[(-1)^(5/6)*x], ( -1)^(2/3)])/(3*x^3*Sqrt[1 + x^2 + x^4])
Time = 0.45 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2248, 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.
| \(\displaystyle \int \frac {3 x^2+2}{x^4 \left (x^2+1\right ) \sqrt {x^4+x^2+1}} \, dx\) |
| \(\Big \downarrow \) 2248 |
| \(\displaystyle \int \left (\frac {1}{\left (-x^2-1\right ) \sqrt {x^4+x^2+1}}+\frac {1}{x^2 \sqrt {x^4+x^2+1}}+\frac {2}{x^4 \sqrt {x^4+x^2+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{3 \sqrt {x^4+x^2+1}}-\frac {\sqrt {x^4+x^2+1} x}{3 \left (x^2+1\right )}+\frac {\sqrt {x^4+x^2+1}}{3 x}-\frac {2 \sqrt {x^4+x^2+1}}{3 x^3}\) |
Input:
Int[(2 + 3*x^2)/(x^4*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
(-2*Sqrt[1 + x^2 + x^4])/(3*x^3) + Sqrt[1 + x^2 + x^4]/(3*x) - (x*Sqrt[1 + x^2 + x^4])/(3*(1 + x^2)) - ArcTan[x/Sqrt[1 + x^2 + x^4]]/2 + ((1 + x^2)* Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(3*Sqrt[1 + x^2 + x^4]) - (3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2* ArcTan[x], 1/4])/(4*Sqrt[1 + x^2 + x^4])
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && In tegerQ[q]
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.88
| method | result | size |
| default | \(\frac {\sqrt {x^{4}+x^{2}+1}}{3 x}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {x^{4}+x^{2}+1}}{3 x^{3}}-\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(339\) |
| risch | \(\frac {x^{6}-x^{4}-x^{2}-2}{3 x^{3} \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(339\) |
| elliptic | \(-\frac {2 \sqrt {x^{4}+x^{2}+1}}{3 x^{3}}+\frac {\sqrt {x^{4}+x^{2}+1}}{3 x}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(408\) |
Input:
int((3*x^2+2)/x^4/(x^2+1)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/x*(x^4+x^2+1)^(1/2)+4/3/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2)) *x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)/(1+I*3^(1 /2))*(EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-E llipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2)))-2/3*(x^ 4+x^2+1)^(1/2)/x^3-4/3/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2) ^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticF(1/2* x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-1/(-1/2+1/2*I*3^(1/2) )^(1/2)*(1+1/2*x^2-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^ (1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I*3^(1/2))^(1/2)*x,-1/(-1/2+1 /2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2))^(1/2))
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {6 \, x^{3} \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) - 2 \, {\left (\sqrt {-3} x^{3} - x^{3}\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - {\left (5 \, \sqrt {-3} x^{3} + 9 \, x^{3}\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 4 \, \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 2\right )}}{12 \, x^{3}} \] Input:
integrate((3*x^2+2)/x^4/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")
Output:
-1/12*(6*x^3*arctan(x/sqrt(x^4 + x^2 + 1)) - 2*(sqrt(-3)*x^3 - x^3)*sqrt(1 /2*sqrt(-3) - 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(-3) - 1/2)), 1/2*sqrt (-3) - 1/2) - (5*sqrt(-3)*x^3 + 9*x^3)*sqrt(1/2*sqrt(-3) - 1/2)*elliptic_f (arcsin(x*sqrt(1/2*sqrt(-3) - 1/2)), 1/2*sqrt(-3) - 1/2) - 4*sqrt(x^4 + x^ 2 + 1)*(x^2 - 2))/x^3
\[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{x^{4} \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:
integrate((3*x**2+2)/x**4/(x**2+1)/(x**4+x**2+1)**(1/2),x)
Output:
Integral((3*x**2 + 2)/(x**4*sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1) ), x)
\[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} x^{4}} \,d x } \] Input:
integrate((3*x^2+2)/x^4/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x^2 + 2)/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)*x^4), x)
\[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} x^{4}} \,d x } \] Input:
integrate((3*x^2+2)/x^4/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate((3*x^2 + 2)/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)*x^4), x)
Timed out. \[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{x^4\,\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \] Input:
int((3*x^2 + 2)/(x^4*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)),x)
Output:
int((3*x^2 + 2)/(x^4*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)), x)
\[ \int \frac {2+3 x^2}{x^4 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {-2 \sqrt {x^{4}+x^{2}+1}-\left (\int \frac {\sqrt {x^{4}+x^{2}+1}}{x^{8}+2 x^{6}+2 x^{4}+x^{2}}d x \right ) x^{3}-6 \left (\int \frac {\sqrt {x^{4}+x^{2}+1}}{x^{6}+2 x^{4}+2 x^{2}+1}d x \right ) x^{3}-2 \left (\int \frac {\sqrt {x^{4}+x^{2}+1}\, x^{2}}{x^{6}+2 x^{4}+2 x^{2}+1}d x \right ) x^{3}}{3 x^{3}} \] Input:
int((3*x^2+2)/x^4/(x^2+1)/(x^4+x^2+1)^(1/2),x)
Output:
( - 2*sqrt(x**4 + x**2 + 1) - int(sqrt(x**4 + x**2 + 1)/(x**8 + 2*x**6 + 2 *x**4 + x**2),x)*x**3 - 6*int(sqrt(x**4 + x**2 + 1)/(x**6 + 2*x**4 + 2*x** 2 + 1),x)*x**3 - 2*int((sqrt(x**4 + x**2 + 1)*x**2)/(x**6 + 2*x**4 + 2*x** 2 + 1),x)*x**3)/(3*x**3)