\(\int \frac {3-x^2}{(1-x^2) \sqrt [4]{1-6 x^2+x^4}} \, dx\) [1253]

Optimal result

Integrand size = 31, antiderivative size = 91 \[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\arctan \left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )+\text {arctanh}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \]

[Out]

arctan((I+x)/(x^4-6*x^2+1)^(1/4))-arctan((x^4-6*x^2+1)^(1/4)/(-I+x))+arctanh((I+x)/(x^4-6*x^2+1)^(1/4))+arctan
h((x^4-6*x^2+1)^(1/4)/(-I+x))

Rubi [F]

\[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx \]

[In]

Int[(3 - x^2)/((1 - x^2)*(1 - 6*x^2 + x^4)^(1/4)),x]

[Out]

Defer[Int][(3 - x^2)/((1 - x^2)*(1 - 6*x^2 + x^4)^(1/4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 6.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\arctan \left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )+\text {arctanh}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \]

[In]

Integrate[(3 - x^2)/((1 - x^2)*(1 - 6*x^2 + x^4)^(1/4)),x]

[Out]

ArcTan[(I + x)/(1 - 6*x^2 + x^4)^(1/4)] - ArcTan[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)] + ArcTanh[(I + x)/(1 - 6*x^
2 + x^4)^(1/4)] + ArcTanh[(1 - 6*x^2 + x^4)^(1/4)/(-I + x)]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.32 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.55

method	result	size
trager	\(\frac {\ln \left (-\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x +\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+x^{4}-\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x -5 x^{2}}{\left (x -1\right ) \left (1+x \right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (x -1\right ) \left (1+x \right )}\right )}{2}\)	\(232\)

[In]

int((-x^2+3)/(-x^2+1)/(x^4-6*x^2+1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(-((x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4)*x^3+x^4-(x^4-6*x^2+1)^(1/2)-3*(x^4-
6*x^2+1)^(1/4)*x-5*x^2)/(x-1)/(1+x))+1/2*RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*(x^4-6*x^2+1)^(1/2)*x^2+RootOf(_Z
^2+1)*x^4+(x^4-6*x^2+1)^(3/4)*x-(x^4-6*x^2+1)^(1/4)*x^3+RootOf(_Z^2+1)*(x^4-6*x^2+1)^(1/2)-5*RootOf(_Z^2+1)*x^
2+3*(x^4-6*x^2+1)^(1/4)*x)/(x-1)/(1+x))

Fricas [F(-2)]

Exception generated. \[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-x^2+3)/(-x^2+1)/(x^4-6*x^2+1)^(1/4),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

\[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\int \frac {x^{2} - 3}{\sqrt [4]{\left (x^{2} - 2 x - 1\right ) \left (x^{2} + 2 x - 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]

[In]

integrate((-x**2+3)/(-x**2+1)/(x**4-6*x**2+1)**(1/4),x)

[Out]

Integral((x**2 - 3)/(((x**2 - 2*x - 1)*(x**2 + 2*x - 1))**(1/4)*(x - 1)*(x + 1)), x)

Maxima [F]

\[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\int { \frac {x^{2} - 3}{{\left (x^{4} - 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate((-x^2+3)/(-x^2+1)/(x^4-6*x^2+1)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^2 - 3)/((x^4 - 6*x^2 + 1)^(1/4)*(x^2 - 1)), x)

Giac [F]

\[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\int { \frac {x^{2} - 3}{{\left (x^{4} - 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate((-x^2+3)/(-x^2+1)/(x^4-6*x^2+1)^(1/4),x, algorithm="giac")

[Out]

integrate((x^2 - 3)/((x^4 - 6*x^2 + 1)^(1/4)*(x^2 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3-x^2}{\left (1-x^2\right ) \sqrt [4]{1-6 x^2+x^4}} \, dx=\int \frac {x^2-3}{\left (x^2-1\right )\,{\left (x^4-6\,x^2+1\right )}^{1/4}} \,d x \]

[In]

int((x^2 - 3)/((x^2 - 1)*(x^4 - 6*x^2 + 1)^(1/4)),x)

[Out]

int((x^2 - 3)/((x^2 - 1)*(x^4 - 6*x^2 + 1)^(1/4)), x)