\(\int \frac {(1+x^2)^2}{(1-x^2) (1-6 x^2+x^4)^{3/4}} \, dx\) [1317]

Optimal result

Integrand size = 31, antiderivative size = 95 \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\arctan \left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right )-\text {arctanh}\left (\frac {i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{-i+x}\right ) \]

[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]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 6.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^2\right )^2}{\left (1-x^2\right ) \left (1-6 x^2+x^4\right )^{3/4}} \, dx=\arctan \left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\arctan \left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right )-\text {arctanh}\left (\frac {-i+x}{\sqrt [4]{1-6 x^2+x^4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{1-6 x^2+x^4}}{i+x}\right ) \]

[In]

Integrate[(1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)),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 = 3.55 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.45

method	result	size
trager	\(\frac {\ln \left (-\frac {\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x -\sqrt {x^{4}-6 x^{2}+1}\, x^{2}+\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-x^{4}+\sqrt {x^{4}-6 x^{2}+1}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{\left (1+x \right ) \left (x -1\right )}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\left (x^{4}-6 x^{2}+1\right )^{\frac {3}{4}} x +\left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-6 x^{2}+1}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}-6 x^{2}+1\right )^{\frac {1}{4}} x}{\left (x -1\right ) \left (1+x \right )}\right )}{2}\)	\(233\)

[In]

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

[In]

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

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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