3.325 \(\int \frac {1-x^2}{(1+x^2) \sqrt {1+x^2+x^4}} \, dx\)

Optimal. Leaf size=15 \[ \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1698, 203} \[ \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]

Rule 203

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

Rule 1698

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.13, size = 94, normalized size = 6.27 \[ -\frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (\operatorname {EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-2 \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{\sqrt {x^4+x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 15, normalized size = 1.00 \[ \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - x^2)/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 13, normalized size = 0.87 \[ \arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(x/sqrt(x^4 + x^2 + 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.49, size = 18, normalized size = 1.20

method	result	size
elliptic	\(-\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\)	\(18\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x -\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\)	\(39\)
default	\(-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/x*(x^4+x^2+1)^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ -\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{x^{2} \sqrt {x^{4} + x^{2} + 1} + \sqrt {x^{4} + x^{2} + 1}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt {x^{4} + x^{2} + 1} + \sqrt {x^{4} + x^{2} + 1}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x**2/(x**2*sqrt(x**4 + x**2 + 1) + sqrt(x**4 + x**2 + 1)), x) - Integral(-1/(x**2*sqrt(x**4 + x**2 +
1) + sqrt(x**4 + x**2 + 1)), x)

________________________________________________________________________________________