3.4.20 \(\int \frac {1-x^2}{(1+x^2) \sqrt {1+x^4}} \, dx\) [320]

Optimal. Leaf size=23 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1713, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 209

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

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + 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^4}} \, dx &=\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.13, size = 112, normalized size = 4.87

method	result	size
elliptic	\(-\frac {\arctan \left (\frac {\sqrt {x^{4}+1}\, \sqrt {2}}{2 x}\right ) \sqrt {2}}{2}\)	\(22\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{2}\)	\(34\)
default	\(-\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 1.62, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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