\(\int \frac {-1+x^2}{(1+x^2) \sqrt {1+x^4}} \, dx\) [241]

Optimal result

Integrand size = 22, antiderivative size = 24 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1713, 209} \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]

[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*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.96 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{2}\)	\(19\)
pseudoelliptic	\(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{2}\)	\(19\)
elliptic	\(\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right ) \sqrt {2}}{2}\)	\(22\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{2}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) \]

[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))

Sympy [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+1}} \,d x \]

[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)