\(\int \frac {3+x^2}{1+3 x^2+x^4} \, dx\) [97]

Optimal result

Integrand size = 18, antiderivative size = 74 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=-\frac {1}{10} \sqrt {180-80 \sqrt {5}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}} \]

[Out]

1/20*arctan(x*(1/2+1/2*5^(1/2)))*(3+5^(1/2))^(3/2)*10^(1/2)-1/10*arctan(x*2^(1/2)/(3+5^(1/2))^(1/2))*(10-4*5^(
1/2))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1180, 209} \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=\frac {\left (3+\sqrt {5}\right )^{3/2} \arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}}-\frac {1}{10} \sqrt {180-80 \sqrt {5}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right ) \]

[In]

Int[(3 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

-1/10*(Sqrt[180 - 80*Sqrt[5]]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + ((3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5]
)/2]*x])/(2*Sqrt[10])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx \\ & = -\frac {1}{5} \sqrt {45-20 \sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=\frac {-\left (3-\sqrt {5}\right )^{3/2} \arctan \left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )+\left (3+\sqrt {5}\right )^{3/2} \arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {10}} \]

[In]

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

[Out]

(-((3 - Sqrt[5])^(3/2)*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + (3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x])
/(2*Sqrt[10])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\left (\textit {\_R}^{2}+3\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}}\right )}{2}\)	\(40\)
default	\(\frac {2 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 x}{2 \sqrt {5}-2}\right )}{5 \left (2 \sqrt {5}-2\right )}+\frac {2 \sqrt {5}\, \left (\sqrt {5}-3\right ) \arctan \left (\frac {4 x}{2 \sqrt {5}+2}\right )}{5 \left (2 \sqrt {5}+2\right )}\)	\(66\)

[In]

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

[Out]

1/2*sum((_R^2+3)/(2*_R^3+3*_R)*ln(x-_R),_R=RootOf(_Z^4+3*_Z^2+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (44) = 88\).

Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.99 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=-\frac {1}{10} \, \sqrt {5} \sqrt {4 \, \sqrt {5} - 9} \log \left (\sqrt {4 \, \sqrt {5} - 9} {\left (3 \, \sqrt {5} + 7\right )} + 2 \, x\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {4 \, \sqrt {5} - 9} \log \left (-\sqrt {4 \, \sqrt {5} - 9} {\left (3 \, \sqrt {5} + 7\right )} + 2 \, x\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {-4 \, \sqrt {5} - 9} \log \left ({\left (3 \, \sqrt {5} - 7\right )} \sqrt {-4 \, \sqrt {5} - 9} + 2 \, x\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {-4 \, \sqrt {5} - 9} \log \left (-{\left (3 \, \sqrt {5} - 7\right )} \sqrt {-4 \, \sqrt {5} - 9} + 2 \, x\right ) \]

[In]

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

[Out]

-1/10*sqrt(5)*sqrt(4*sqrt(5) - 9)*log(sqrt(4*sqrt(5) - 9)*(3*sqrt(5) + 7) + 2*x) + 1/10*sqrt(5)*sqrt(4*sqrt(5)
- 9)*log(-sqrt(4*sqrt(5) - 9)*(3*sqrt(5) + 7) + 2*x) - 1/10*sqrt(5)*sqrt(-4*sqrt(5) - 9)*log((3*sqrt(5) - 7)*
sqrt(-4*sqrt(5) - 9) + 2*x) + 1/10*sqrt(5)*sqrt(-4*sqrt(5) - 9)*log(-(3*sqrt(5) - 7)*sqrt(-4*sqrt(5) - 9) + 2*
x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.62 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=2 \left (\frac {\sqrt {5}}{5} + \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {2 x}{-1 + \sqrt {5}} \right )} - 2 \cdot \left (\frac {1}{2} - \frac {\sqrt {5}}{5}\right ) \operatorname {atan}{\left (\frac {2 x}{1 + \sqrt {5}} \right )} \]

[In]

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

[Out]

2*(sqrt(5)/5 + 1/2)*atan(2*x/(-1 + sqrt(5))) - 2*(1/2 - sqrt(5)/5)*atan(2*x/(1 + sqrt(5)))

Maxima [F]

\[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=\int { \frac {x^{2} + 3}{x^{4} + 3 \, x^{2} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.55 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=\frac {1}{5} \, {\left (2 \, \sqrt {5} - 5\right )} \arctan \left (\frac {2 \, x}{\sqrt {5} + 1}\right ) + \frac {1}{5} \, {\left (2 \, \sqrt {5} + 5\right )} \arctan \left (\frac {2 \, x}{\sqrt {5} - 1}\right ) \]

[In]

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

[Out]

1/5*(2*sqrt(5) - 5)*arctan(2*x/(sqrt(5) + 1)) + 1/5*(2*sqrt(5) + 5)*arctan(2*x/(sqrt(5) - 1))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.58 \[ \int \frac {3+x^2}{1+3 x^2+x^4} \, dx=2\,\mathrm {atanh}\left (\frac {80\,x\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}-56}-\frac {48\,\sqrt {5}\,x\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}-56}\right )\,\sqrt {\frac {\sqrt {5}}{5}-\frac {9}{20}}-2\,\mathrm {atanh}\left (\frac {80\,x\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}+56}+\frac {48\,\sqrt {5}\,x\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}}}{24\,\sqrt {5}+56}\right )\,\sqrt {-\frac {\sqrt {5}}{5}-\frac {9}{20}} \]

[In]

int((x^2 + 3)/(3*x^2 + x^4 + 1),x)

[Out]

2*atanh((80*x*(5^(1/2)/5 - 9/20)^(1/2))/(24*5^(1/2) - 56) - (48*5^(1/2)*x*(5^(1/2)/5 - 9/20)^(1/2))/(24*5^(1/2
) - 56))*(5^(1/2)/5 - 9/20)^(1/2) - 2*atanh((80*x*(- 5^(1/2)/5 - 9/20)^(1/2))/(24*5^(1/2) + 56) + (48*5^(1/2)*
x*(- 5^(1/2)/5 - 9/20)^(1/2))/(24*5^(1/2) + 56))*(- 5^(1/2)/5 - 9/20)^(1/2)