\(\int \frac {-1+x^4}{(1+x^4) \sqrt {1+x^2+x^4}} \, dx\) [118]

Optimal result

Integrand size = 25, antiderivative size = 17 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]

[Out]

-arctanh(x/(x^4+x^2+1)^(1/2))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2137, 212} \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + x^2 + x^4]]

Rule 212

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

Rule 2137

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + x^2 + x^4]]

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
elliptic	\(-\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\)	\(18\)
trager	\(\frac {\ln \left (-\frac {-x^{4}+2 x \sqrt {x^{4}+x^{2}+1}-2 x^{2}-1}{x^{4}+1}\right )}{2}\)	\(38\)
default	\(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}\)	\(56\)
pseudoelliptic	\(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )}{2}\)	\(56\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + 1} x + 1}{x^{4} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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