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

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

[Out]

ArcTan[x/Sqrt[1 + x^2 + x^4]]

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Rubi [A]  time = 0.0390042, antiderivative size = 15, normalized size of antiderivative = 1., 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 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]

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])

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 = 0.129821, size = 94, normalized size = 6.27 \[ -\frac{(-1)^{2/3} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \left (\text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+2 \Pi \left (\sqrt [3]{-1};-i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{\sqrt{x^4+x^2+1}} \]

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])

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Maple [C]  time = 0.023, size = 188, normalized size = 12.5 \begin{align*} -2\,{\frac{\sqrt{1- \left ( -1/2+i/2\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -1/2-i/2\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-2+2\,i\sqrt{3}},1/2\,\sqrt{-2+2\,i\sqrt{3}} \right ) }{\sqrt{-2+2\,i\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}}+2\,{\frac{\sqrt{1+1/2\,{x}^{2}-i/2{x}^{2}\sqrt{3}}\sqrt{1+1/2\,{x}^{2}+i/2{x}^{2}\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}\sqrt{{x}^{4}+{x}^{2}+1}}{\it EllipticPi} \left ( \sqrt{-1/2+i/2\sqrt{3}}x,- \left ( -1/2+i/2\sqrt{3} \right ) ^{-1},{\frac{\sqrt{-1/2-i/2\sqrt{3}}}{\sqrt{-1/2+i/2\sqrt{3}}}} \right ) } \end{align*}

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)

[Out]

-2/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1
/2)*EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))+2/(-1/2+1/2*I*3^(1/2))^(1/2)*(1+1/2*x^2
-1/2*I*x^2*3^(1/2))^(1/2)*(1+1/2*x^2+1/2*I*x^2*3^(1/2))^(1/2)/(x^4+x^2+1)^(1/2)*EllipticPi((-1/2+1/2*I*3^(1/2)
)^(1/2)*x,-1/(-1/2+1/2*I*3^(1/2)),(-1/2-1/2*I*3^(1/2))^(1/2)/(-1/2+1/2*I*3^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}

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)

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Fricas [A]  time = 1.7684, size = 42, normalized size = 2.8 \begin{align*} \arctan \left (\frac{x}{\sqrt{x^{4} + x^{2} + 1}}\right ) \end{align*}

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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + x^{2} + 1} + \sqrt{x^{4} + x^{2} + 1}}\, dx - \int - \frac{1}{x^{2} \sqrt{x^{4} + x^{2} + 1} + \sqrt{x^{4} + x^{2} + 1}}\, dx \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} - 1}{\sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}

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)