\(\int \frac {1+x^4}{(1-x^4) \sqrt {-1+x^2+x^4}} \, dx\) [68]

Optimal result

Integrand size = 27, antiderivative size = 15 \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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 = 0.21 (sec) , antiderivative size = 15, 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 = 3.76 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
elliptic	\(\operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\)	\(16\)
trager	\(\frac {\ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+x^{2}-1}+2 x^{2}-1}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(46\)
default	\(-\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}\)	\(62\)
pseudoelliptic	\(-\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (-1+2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {\left (1+i\right ) x^{2}+\left (1-2 i\right ) x -1+i}{\sqrt {x^{4}+x^{2}-1}}\right )}{2}\)	\(62\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \frac {1+x^4}{\left (1-x^4\right ) \sqrt {-1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} - 1} x - 1}{x^{4} - 1}\right ) \]

[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]

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

[In]

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

[Out]

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