\(\int \frac {2+3 x^2}{x^2 (1+x^2) \sqrt {1+x^2+x^4}} \, dx\) [102]

Optimal result

Integrand size = 30, antiderivative size = 155 \[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {2 \sqrt {1+x^2+x^4}}{x}+\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {1}{2} \arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right )-\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {1+x^2+x^4}}+\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}} \] Output:

-2*(x^4+x^2+1)^(1/2)/x+2*x*(x^4+x^2+1)^(1/2)/(x^2+1)+1/2*arctan(x/(x^4+x^2 
+1)^(1/2))-2*(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)+5/4*(x^2+1)*((x^4+x^2+1)/(x^2+1)^2)^(1/2)*Inver 
seJacobiAM(2*arctan(x),1/2)/(x^4+x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06 \[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {-\frac {2 \left (1+x^2+x^4\right )}{x}+2 \sqrt [3]{-1} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \left (E\left (i \text {arcsinh}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )\right )+(-1)^{2/3} \sqrt {1+\sqrt [3]{-1} x^2} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticPi}\left (\sqrt [3]{-1},i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{\sqrt {1+x^2+x^4}} \] Input:

Integrate[(2 + 3*x^2)/(x^2*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
 

Output:

((-2*(1 + x^2 + x^4))/x + 2*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - ( 
-1)^(2/3)*x^2]*(EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - EllipticF 
[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]) + (-1)^(2/3)*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)])/Sqrt[1 + x^2 + x^4]
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 155, 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.

\(\displaystyle \int \frac {3 x^2+2}{x^2 \left (x^2+1\right ) \sqrt {x^4+x^2+1}} \, dx\)

\(\Big \downarrow \) 2248

\(\displaystyle \int \left (\frac {1}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}+\frac {2}{x^2 \sqrt {x^4+x^2+1}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right )+\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{4 \sqrt {x^4+x^2+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \arctan (x)\left |\frac {1}{4}\right .\right )}{\sqrt {x^4+x^2+1}}+\frac {2 \sqrt {x^4+x^2+1} x}{x^2+1}-\frac {2 \sqrt {x^4+x^2+1}}{x}\)

Input:

Int[(2 + 3*x^2)/(x^2*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
 

Output:

(-2*Sqrt[1 + x^2 + x^4])/x + (2*x*Sqrt[1 + x^2 + x^4])/(1 + x^2) + ArcTan[ 
x/Sqrt[1 + x^2 + x^4]]/2 - (2*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]* 
EllipticE[2*ArcTan[x], 1/4])/Sqrt[1 + x^2 + x^4] + (5*(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 
])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2248

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \sqrt {x^{4}+x^{2}+1}}{x}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(241\)
risch	\(\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \sqrt {x^{4}+x^{2}+1}}{x}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}\)	\(241\)
elliptic	\(-\frac {2 \sqrt {x^{4}+x^{2}+1}}{x}-\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(308\)

Input:

int((3*x^2+2)/x^2/(x^2+1)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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))-2/x*(x^4+x^2+1)^(1/2)-8/(-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))-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/ 
2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.74 \[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {4 \, {\left (\sqrt {-3} x - x\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - {\left (3 \, \sqrt {-3} x - 5 \, x\right )} \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}}\right )\,|\,\frac {1}{2} \, \sqrt {-3} - \frac {1}{2}) - 2 \, x \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) + 8 \, \sqrt {x^{4} + x^{2} + 1}}{4 \, x} \] Input:

integrate((3*x^2+2)/x^2/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/4*(4*(sqrt(-3)*x - x)*sqrt(1/2*sqrt(-3) - 1/2)*elliptic_e(arcsin(x*sqrt 
(1/2*sqrt(-3) - 1/2)), 1/2*sqrt(-3) - 1/2) - (3*sqrt(-3)*x - 5*x)*sqrt(1/2 
*sqrt(-3) - 1/2)*elliptic_f(arcsin(x*sqrt(1/2*sqrt(-3) - 1/2)), 1/2*sqrt(- 
3) - 1/2) - 2*x*arctan(x/sqrt(x^4 + x^2 + 1)) + 8*sqrt(x^4 + x^2 + 1))/x
 

Sympy [F]

\[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{x^{2} \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:

integrate((3*x**2+2)/x**2/(x**2+1)/(x**4+x**2+1)**(1/2),x)
 

Output:

Integral((3*x**2 + 2)/(x**2*sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1) 
), x)
 

Maxima [F]

\[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:

integrate((3*x^2+2)/x^2/(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^2), x)
 

Giac [F]

\[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} x^{2}} \,d x } \] Input:

integrate((3*x^2+2)/x^2/(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^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{x^2\,\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \] Input:

int((3*x^2 + 2)/(x^2*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)),x)
 

Output:

int((3*x^2 + 2)/(x^2*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {2+3 x^2}{x^2 \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=2 \left (\int \frac {\sqrt {x^{4}+x^{2}+1}}{x^{8}+2 x^{6}+2 x^{4}+x^{2}}d x \right )+3 \left (\int \frac {\sqrt {x^{4}+x^{2}+1}}{x^{6}+2 x^{4}+2 x^{2}+1}d x \right ) \] Input:

int((3*x^2+2)/x^2/(x^2+1)/(x^4+x^2+1)^(1/2),x)
 

Output:

2*int(sqrt(x**4 + x**2 + 1)/(x**8 + 2*x**6 + 2*x**4 + x**2),x) + 3*int(sqr 
t(x**4 + x**2 + 1)/(x**6 + 2*x**4 + 2*x**2 + 1),x)