3.10.18 \(\int \frac {x^2}{1-x^2+x^4} \, dx\) [918]

Optimal. Leaf size=74 \[ -\frac {1}{2} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}} \]

[Out]

1/2*arctan(2*x-3^(1/2))+1/2*arctan(2*x+3^(1/2))+1/12*ln(1+x^2-x*3^(1/2))*3^(1/2)-1/12*ln(1+x^2+x*3^(1/2))*3^(1
/2)

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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1141, 1175, 632, 210, 1178, 642} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\sqrt {3}-2 x\right )+\frac {1}{2} \text {ArcTan}\left (2 x+\sqrt {3}\right )+\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(1 - x^2 + x^4),x]

[Out]

-1/2*ArcTan[Sqrt[3] - 2*x] + ArcTan[Sqrt[3] + 2*x]/2 + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*
x + x^2]/(4*Sqrt[3])

Rule 210

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])

Rule 632

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 642

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 1141

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

Rule 1175

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

Rule 1178

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 94, normalized size = 1.27 \begin {gather*} \frac {\sqrt {-1-i \sqrt {3}} \left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )+\sqrt {-1+i \sqrt {3}} \left (-i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )}{2 \sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(1 - x^2 + x^4),x]

[Out]

(Sqrt[-1 - I*Sqrt[3]]*(I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2] + Sqrt[-1 + I*Sqrt[3]]*(-I + Sqrt[3])*ArcTan
[((1 + I*Sqrt[3])*x)/2])/(2*Sqrt[6])

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Maple [A]
time = 0.03, size = 69, normalized size = 0.93

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-\textit {\_R}}\right )}{2}\)	\(38\)
default	\(-\frac {\sqrt {3}\, \left (-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x -\sqrt {3}\right )\right )}{6}-\frac {\sqrt {3}\, \left (\frac {\ln \left (1+x^{2}+x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x +\sqrt {3}\right )\right )}{6}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*3^(1/2)*(-1/2*ln(1+x^2-x*3^(1/2))-3^(1/2)*arctan(2*x-3^(1/2)))-1/6*3^(1/2)*(1/2*ln(1+x^2+x*3^(1/2))-3^(1/
2)*arctan(2*x+3^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(x^4 - x^2 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (56) = 112\).
time = 0.39, size = 163, normalized size = 2.20 \begin {gather*} -\frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{18} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36} + \sqrt {3}\right ) - \frac {1}{6} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2} - \sqrt {3}\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {2} \log \left (18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \log \left (-18 \, \sqrt {6} \sqrt {2} x + 36 \, x^{2} + 36\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(6)*sqrt(3)*sqrt(2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x + 1/18*sqrt(6)*sqrt(3)*sqrt(2)*sqrt(-18*sqr
t(6)*sqrt(2)*x + 36*x^2 + 36) + sqrt(3)) - 1/6*sqrt(6)*sqrt(3)*sqrt(2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x +
1/3*sqrt(6)*sqrt(3)*sqrt(sqrt(6)*sqrt(2)*x + 2*x^2 + 2) - sqrt(3)) - 1/24*sqrt(6)*sqrt(2)*log(18*sqrt(6)*sqrt
(2)*x + 36*x^2 + 36) + 1/24*sqrt(6)*sqrt(2)*log(-18*sqrt(6)*sqrt(2)*x + 36*x^2 + 36)

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Sympy [A]
time = 0.07, size = 63, normalized size = 0.85 \begin {gather*} \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{2} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + atan(2*x - sqrt(3))/2 + atan(2*x
+ sqrt(3))/2

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Giac [A]
time = 3.70, size = 56, normalized size = 0.76 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{2} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{2} \, \arctan \left (2 \, x - \sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + 1/2*arctan(2*x + sqrt(3)) + 1
/2*arctan(2*x - sqrt(3))

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Mupad [B]
time = 0.08, size = 44, normalized size = 0.59 \begin {gather*} -\mathrm {atan}\left (\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atan}\left (\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4 - x^2 + 1),x)

[Out]

atan(x/2 + (3^(1/2)*x*1i)/2)*((3^(1/2)*1i)/6 + 1/2) - atan(x/2 - (3^(1/2)*x*1i)/2)*((3^(1/2)*1i)/6 - 1/2)

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