\(\int \frac {(1-x^2)^2}{(1+x^2) (1+6 x^2+x^4)^{3/4}} \, dx\) [1079]

Optimal result

Integrand size = 31, antiderivative size = 81 \[ \int \frac {\left (1-x^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx=-\arctan \left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )-\arctan \left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right ) \]

[Out]

-arctan((-1+x)/(x^4+6*x^2+1)^(1/4))-arctan((1+x)/(x^4+6*x^2+1)^(1/4))+arctanh((-1+x)/(x^4+6*x^2+1)^(1/4))+arct
anh((1+x)/(x^4+6*x^2+1)^(1/4))

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 = 6.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^2\right )^2}{\left (1+x^2\right ) \left (1+6 x^2+x^4\right )^{3/4}} \, dx=-\arctan \left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )-\arctan \left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {-1+x}{\sqrt [4]{1+6 x^2+x^4}}\right )+\text {arctanh}\left (\frac {1+x}{\sqrt [4]{1+6 x^2+x^4}}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 3.70 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.68

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\sqrt {x^{4}+6 x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{3} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{3} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{5}-\sqrt {x^{4}+6 x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{3} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{3} x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )^{2} \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x -\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}+6 x^{2}+1}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x}{{\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +1\right )}^{2} {\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -1\right )}^{2}}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}+6 x^{2}+1\right )^{\frac {3}{4}} x +x^{2} \sqrt {x^{4}+6 x^{2}+1}+\left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x^{3}+x^{4}+\sqrt {x^{4}+6 x^{2}+1}+3 \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}} x +5 x^{2}}{x^{2}+1}\right )}{2}\)	\(379\)

[In]

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

[Out]

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

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]

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

[In]

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

[Out]

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

[Out]

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