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

Optimal. Leaf size=17 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1698, 203} \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}

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*}

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Mathematica [C]  time = 0.13, size = 93, normalized size = 5.47 \begin {gather*} \frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (F\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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.17, size = 17, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \end {gather*}

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]]

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fricas [A]  time = 0.50, size = 15, normalized size = 0.88 \begin {gather*} -\arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \end {gather*}

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

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)

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maple [A]  time = 0.42, size = 16, normalized size = 0.94

method	result	size
elliptic	\(\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\)	\(16\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\)	\(36\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\)	\(188\)

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))

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}

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)

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

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

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