\(\int \frac {1+x^2}{(1-x^2) \sqrt {1+x^4}} \, dx\) [321]

Optimal result

Integrand size = 24, antiderivative size = 23 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1713, 212} \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

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 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 {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
elliptic	\(\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right ) \sqrt {2}}{2}\)	\(22\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}\)	\(38\)
default	\(\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{4}\)	\(47\)
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{4}\)	\(47\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[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)

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)