Integrand size = 12, antiderivative size = 57 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=-\frac {\arctan \left (\frac {1-2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}+\frac {\arctan \left (\frac {1+2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}+\frac {1}{6} \text {arctanh}\left (\frac {x}{3+x^2}\right ) \] Output:
-1/66*arctan(1/11*(1-2*x)*11^(1/2))*11^(1/2)+1/66*arctan(1/11*(1+2*x)*11^( 1/2))*11^(1/2)+1/6*arctanh(x/(x^2+3))
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.60 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=-\frac {i \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (5-i \sqrt {11}\right )}}\right )}{\sqrt {\frac {11}{2} \left (5-i \sqrt {11}\right )}}+\frac {i \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (5+i \sqrt {11}\right )}}\right )}{\sqrt {\frac {11}{2} \left (5+i \sqrt {11}\right )}} \] Input:
Integrate[(9 + 5*x^2 + x^4)^(-1),x]
Output:
((-I)*ArcTan[x/Sqrt[(5 - I*Sqrt[11])/2]])/Sqrt[(11*(5 - I*Sqrt[11]))/2] + (I*ArcTan[x/Sqrt[(5 + I*Sqrt[11])/2]])/Sqrt[(11*(5 + I*Sqrt[11]))/2]
Time = 0.40 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1407, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {1}{x^4+5 x^2+9} \, dx\) |
\(\Big \downarrow \) 1407 |
\(\displaystyle \frac {1}{6} \int \frac {1-x}{x^2-x+3}dx+\frac {1}{6} \int \frac {x+1}{x^2+x+3}dx\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2-x+3}dx-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+3}dx\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2+x+3}dx+\frac {1}{2} \int \frac {2 x+1}{x^2+x+3}dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2-x+3}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+3}dx\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2+x+3}dx+\frac {1}{2} \int \frac {2 x+1}{x^2+x+3}dx\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+3}dx-\int \frac {1}{-(2 x-1)^2-11}d(2 x-1)\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 x+1}{x^2+x+3}dx-\int \frac {1}{-(2 x+1)^2-11}d(2 x+1)\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+3}dx+\frac {\arctan \left (\frac {2 x-1}{\sqrt {11}}\right )}{\sqrt {11}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 x+1}{x^2+x+3}dx+\frac {\arctan \left (\frac {2 x+1}{\sqrt {11}}\right )}{\sqrt {11}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{6} \left (\frac {\arctan \left (\frac {2 x-1}{\sqrt {11}}\right )}{\sqrt {11}}-\frac {1}{2} \log \left (x^2-x+3\right )\right )+\frac {1}{6} \left (\frac {\arctan \left (\frac {2 x+1}{\sqrt {11}}\right )}{\sqrt {11}}+\frac {1}{2} \log \left (x^2+x+3\right )\right )\) |
Input:
Int[(9 + 5*x^2 + x^4)^(-1),x]
Output:
(ArcTan[(-1 + 2*x)/Sqrt[11]]/Sqrt[11] - Log[3 - x + x^2]/2)/6 + (ArcTan[(1 + 2*x)/Sqrt[11]]/Sqrt[11] + Log[3 + x + x^2]/2)/6
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {\ln \left (x^{2}-x +3\right )}{12}+\frac {\sqrt {11}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {11}}{11}\right )}{66}+\frac {\ln \left (x^{2}+x +3\right )}{12}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {11}}{11}\right ) \sqrt {11}}{66}\) | \(54\) |
risch | \(-\frac {\ln \left (4 x^{2}-4 x +12\right )}{12}+\frac {\sqrt {11}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {11}}{11}\right )}{66}+\frac {\ln \left (4 x^{2}+4 x +12\right )}{12}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {11}}{11}\right ) \sqrt {11}}{66}\) | \(60\) |
Input:
int(1/(x^4+5*x^2+9),x,method=_RETURNVERBOSE)
Output:
-1/12*ln(x^2-x+3)+1/66*11^(1/2)*arctan(1/11*(2*x-1)*11^(1/2))+1/12*ln(x^2+ x+3)+1/66*arctan(1/11*(1+2*x)*11^(1/2))*11^(1/2)
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=\frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 1\right )}\right ) + \frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} + x + 3\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 3\right ) \] Input:
integrate(1/(x^4+5*x^2+9),x, algorithm="fricas")
Output:
1/66*sqrt(11)*arctan(1/11*sqrt(11)*(2*x + 1)) + 1/66*sqrt(11)*arctan(1/11* sqrt(11)*(2*x - 1)) + 1/12*log(x^2 + x + 3) - 1/12*log(x^2 - x + 3)
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=- \frac {\log {\left (x^{2} - x + 3 \right )}}{12} + \frac {\log {\left (x^{2} + x + 3 \right )}}{12} + \frac {\sqrt {11} \operatorname {atan}{\left (\frac {2 \sqrt {11} x}{11} - \frac {\sqrt {11}}{11} \right )}}{66} + \frac {\sqrt {11} \operatorname {atan}{\left (\frac {2 \sqrt {11} x}{11} + \frac {\sqrt {11}}{11} \right )}}{66} \] Input:
integrate(1/(x**4+5*x**2+9),x)
Output:
-log(x**2 - x + 3)/12 + log(x**2 + x + 3)/12 + sqrt(11)*atan(2*sqrt(11)*x/ 11 - sqrt(11)/11)/66 + sqrt(11)*atan(2*sqrt(11)*x/11 + sqrt(11)/11)/66
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=\frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 1\right )}\right ) + \frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} + x + 3\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 3\right ) \] Input:
integrate(1/(x^4+5*x^2+9),x, algorithm="maxima")
Output:
1/66*sqrt(11)*arctan(1/11*sqrt(11)*(2*x + 1)) + 1/66*sqrt(11)*arctan(1/11* sqrt(11)*(2*x - 1)) + 1/12*log(x^2 + x + 3) - 1/12*log(x^2 - x + 3)
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=\frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 1\right )}\right ) + \frac {1}{66} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} + x + 3\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 3\right ) \] Input:
integrate(1/(x^4+5*x^2+9),x, algorithm="giac")
Output:
1/66*sqrt(11)*arctan(1/11*sqrt(11)*(2*x + 1)) + 1/66*sqrt(11)*arctan(1/11* sqrt(11)*(2*x - 1)) + 1/12*log(x^2 + x + 3) - 1/12*log(x^2 - x + 3)
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.46 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=\mathrm {atan}\left (\frac {x\,8{}\mathrm {i}}{27\,\left (-\frac {5}{9}+\frac {\sqrt {11}\,1{}\mathrm {i}}{9}\right )}-\frac {2\,\sqrt {11}\,x}{27\,\left (-\frac {5}{9}+\frac {\sqrt {11}\,1{}\mathrm {i}}{9}\right )}\right )\,\left (\frac {\sqrt {11}}{66}+\frac {1}{6}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {x\,8{}\mathrm {i}}{27\,\left (\frac {5}{9}+\frac {\sqrt {11}\,1{}\mathrm {i}}{9}\right )}+\frac {2\,\sqrt {11}\,x}{27\,\left (\frac {5}{9}+\frac {\sqrt {11}\,1{}\mathrm {i}}{9}\right )}\right )\,\left (\frac {\sqrt {11}}{66}-\frac {1}{6}{}\mathrm {i}\right ) \] Input:
int(1/(5*x^2 + x^4 + 9),x)
Output:
atan((x*8i)/(27*((11^(1/2)*1i)/9 - 5/9)) - (2*11^(1/2)*x)/(27*((11^(1/2)*1 i)/9 - 5/9)))*(11^(1/2)/66 + 1i/6) + atan((x*8i)/(27*((11^(1/2)*1i)/9 + 5/ 9)) + (2*11^(1/2)*x)/(27*((11^(1/2)*1i)/9 + 5/9)))*(11^(1/2)/66 - 1i/6)
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {1}{9+5 x^2+x^4} \, dx=\frac {\sqrt {11}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {11}}\right )}{66}+\frac {\sqrt {11}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {11}}\right )}{66}-\frac {\mathrm {log}\left (x^{2}-x +3\right )}{12}+\frac {\mathrm {log}\left (x^{2}+x +3\right )}{12} \] Input:
int(1/(x^4+5*x^2+9),x)
Output:
(2*sqrt(11)*atan((2*x - 1)/sqrt(11)) + 2*sqrt(11)*atan((2*x + 1)/sqrt(11)) - 11*log(x**2 - x + 3) + 11*log(x**2 + x + 3))/132