3.1.13 \(\int \frac {1}{9+5 x^2+x^4} \, dx\)

Optimal. Leaf size=67 \[ -\frac {1}{12} \log \left (x^2-x+3\right )+\frac {1}{12} \log \left (x^2+x+3\right )-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {11}}\right )}{6 \sqrt {11}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1094, 634, 618, 204, 628} \begin {gather*} -\frac {1}{12} \log \left (x^2-x+3\right )+\frac {1}{12} \log \left (x^2+x+3\right )-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {11}}\right )}{6 \sqrt {11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 + 5*x^2 + x^4)^(-1),x]

[Out]

-ArcTan[(1 - 2*x)/Sqrt[11]]/(6*Sqrt[11]) + ArcTan[(1 + 2*x)/Sqrt[11]]/(6*Sqrt[11]) - Log[3 - x + x^2]/12 + Log
[3 + x + x^2]/12

Rule 204

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rule 634

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

Rule 1094

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]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[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]

Rubi steps

\begin {align*} \int \frac {1}{9+5 x^2+x^4} \, dx &=\frac {1}{6} \int \frac {1-x}{3-x+x^2} \, dx+\frac {1}{6} \int \frac {1+x}{3+x+x^2} \, dx\\ &=\frac {1}{12} \int \frac {1}{3-x+x^2} \, dx-\frac {1}{12} \int \frac {-1+2 x}{3-x+x^2} \, dx+\frac {1}{12} \int \frac {1}{3+x+x^2} \, dx+\frac {1}{12} \int \frac {1+2 x}{3+x+x^2} \, dx\\ &=-\frac {1}{12} \log \left (3-x+x^2\right )+\frac {1}{12} \log \left (3+x+x^2\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {11}}\right )}{6 \sqrt {11}}-\frac {1}{12} \log \left (3-x+x^2\right )+\frac {1}{12} \log \left (3+x+x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.07, size = 91, normalized size = 1.36 \begin {gather*} \frac {i \tan ^{-1}\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 \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (5-i \sqrt {11}\right )}}\right )}{\sqrt {\frac {11}{2} \left (5-i \sqrt {11}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 5*x^2 + x^4)^(-1),x]

[Out]

((-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]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{9+5 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(9 + 5*x^2 + x^4)^(-1),x]

[Out]

IntegrateAlgebraic[(9 + 5*x^2 + x^4)^(-1), x]

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 53, normalized size = 0.79 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+5*x^2+9),x, algorithm="fricas")

[Out]

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)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 53, normalized size = 0.79 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+5*x^2+9),x, algorithm="giac")

[Out]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 54, normalized size = 0.81 \begin {gather*} \frac {\sqrt {11}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {11}}{11}\right )}{66}+\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 {\ln \left (x^{2}+x +3\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4+5*x^2+9),x)

[Out]

-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*(2*x+1)*11^(1/
2))*11^(1/2)

________________________________________________________________________________________

maxima [A]  time = 3.04, size = 53, normalized size = 0.79 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+5*x^2+9),x, algorithm="maxima")

[Out]

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)

________________________________________________________________________________________

mupad [B]  time = 4.15, size = 83, normalized size = 1.24 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2 + x^4 + 9),x)

[Out]

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) +
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)

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 70, normalized size = 1.04 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4+5*x**2+9),x)

[Out]

-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

________________________________________________________________________________________