\(\int \frac {(-1+x^4) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx\) [1256]

Optimal result

Integrand size = 33, antiderivative size = 91 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=-\sqrt {\frac {1}{6} \left (-1-i \sqrt {3}\right )} \arctan \left (\frac {2 x}{\left (-i+\sqrt {3}\right ) \sqrt {1+x^4}}\right )-\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} \arctan \left (\frac {2 x}{\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}\right ) \]

[Out]

-1/6*(-6-6*I*3^(1/2))^(1/2)*arctan(2*x/(-I+3^(1/2))/(x^4+1)^(1/2))-1/6*(-6+6*I*3^(1/2))^(1/2)*arctan(2*x/(3^(1
/2)+I)/(x^4+1)^(1/2))

Rubi [F]

\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx \]

[In]

Int[((-1 + x^4)*Sqrt[1 + x^4])/(1 + x^2 + 3*x^4 + x^6 + x^8),x]

[Out]

Defer[Int][Sqrt[1 + x^4]/(-1 - x^2 - 3*x^4 - x^6 - x^8), x] + Defer[Int][(x^4*Sqrt[1 + x^4])/(1 + x^2 + 3*x^4
+ x^6 + x^8), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {1+x^4}}{-1-x^2-3 x^4-x^6-x^8}+\frac {x^4 \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8}\right ) \, dx \\ & = \int \frac {\sqrt {1+x^4}}{-1-x^2-3 x^4-x^6-x^8} \, dx+\int \frac {x^4 \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x \sqrt {1+x^4}}{1-x^2+x^4}\right )}{2 \sqrt {3}}-\frac {1}{2} \text {arctanh}\left (\frac {x \sqrt {1+x^4}}{1+x^2+x^4}\right ) \]

[In]

Integrate[((-1 + x^4)*Sqrt[1 + x^4])/(1 + x^2 + 3*x^4 + x^6 + x^8),x]

[Out]

-1/2*ArcTan[(Sqrt[3]*x*Sqrt[1 + x^4])/(1 - x^2 + x^4)]/Sqrt[3] - ArcTanh[(x*Sqrt[1 + x^4])/(1 + x^2 + x^4)]/2

Maple [A] (verified)

Time = 4.74 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\right ) \sqrt {2}}{2}\)	\(126\)
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}+\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+1}{x^{2}}-\frac {\sqrt {x^{4}+1}}{x}+1\right )}{4}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {2}\, \sqrt {x^{4}+1}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\right ) \sqrt {2}}{2}\)	\(126\)
trager	\(\operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (\frac {x^{4}+12 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+4 \sqrt {x^{4}+1}\, x +x^{2}+1}{-x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+x^{2}-1}\right )-\frac {\ln \left (-\frac {-x^{4}+12 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +2 x^{2}-1}{x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 x^{2}+1}\right )}{2}-\ln \left (-\frac {-x^{4}+12 \sqrt {x^{4}+1}\, \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x +6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x +2 x^{2}-1}{x^{4}+6 \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+2 x^{2}+1}\right ) \operatorname {RootOf}\left (12 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )\)	\(294\)
pseudoelliptic	\(-\frac {\ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+1}-i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}-x^{3}-x +2}{x^{2}}\right )}{4}+\frac {i \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+1}-i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}-x^{3}-x +2}{x^{2}}\right ) \sqrt {3}}{12}-\frac {i \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+1}+i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}-x^{3}-x +2}{x^{2}}\right ) \sqrt {3}}{12}-\frac {\ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}+x -2\right ) \sqrt {x^{4}+1}+i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}-x^{3}-x +2}{x^{2}}\right )}{4}+\frac {\ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+1}-i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}+x^{3}+x +2}{x^{2}}\right )}{4}+\frac {i \ln \left (\frac {\left (i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+1}-i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}+x^{3}+x +2}{x^{2}}\right ) \sqrt {3}}{12}+\frac {\ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+1}+i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}+x^{3}+x +2}{x^{2}}\right )}{4}-\frac {i \ln \left (\frac {\left (-i \sqrt {3}\, x -2 x^{2}-x -2\right ) \sqrt {x^{4}+1}+i \left (x^{2}+1\right ) x \sqrt {3}+2 x^{4}+x^{3}+x +2}{x^{2}}\right ) \sqrt {3}}{12}\)	\(466\)

[In]

int((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/4*2^(1/2)*ln((x^4+1)/x^2+(x^4+1)^(1/2)/x+1)+1/6*6^(1/2)*arctan(1/6*(2*2^(1/2)/x*(x^4+1)^(1/2)+2^(1/2))
*6^(1/2))+1/4*2^(1/2)*ln((x^4+1)/x^2-(x^4+1)^(1/2)/x+1)+1/6*6^(1/2)*arctan(1/6*(2*2^(1/2)/x*(x^4+1)^(1/2)-2^(1
/2))*6^(1/2)))*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{4} - x^{2} + 1\right )} \sqrt {x^{4} + 1}}{3 \, {\left (x^{5} + x\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} - 2 \, {\left (x^{5} + x^{3} + x\right )} \sqrt {x^{4} + 1} + 1}{x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1}\right ) \]

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(x^4 - x^2 + 1)*sqrt(x^4 + 1)/(x^5 + x)) + 1/4*log((x^8 + 3*x^6 + 3*x^4 + 3*x^2
- 2*(x^5 + x^3 + x)*sqrt(x^4 + 1) + 1)/(x^8 + x^6 + 3*x^4 + x^2 + 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\text {Timed out} \]

[In]

integrate((x**4-1)*(x**4+1)**(1/2)/(x**8+x**6+3*x**4+x**2+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\int { \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1} \,d x } \]

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + 1)*(x^4 - 1)/(x^8 + x^6 + 3*x^4 + x^2 + 1), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((x^4-1)*(x^4+1)^(1/2)/(x^8+x^6+3*x^4+x^2+1),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:  ***   Warning:
increasing stack size to 2048000.  ***   Warning: increasing stack size to 2048000.  ***   Warning: increasing
stack size to

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^2+3 x^4+x^6+x^8} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}}{x^8+x^6+3\,x^4+x^2+1} \,d x \]

[In]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(x^2 + 3*x^4 + x^6 + x^8 + 1),x)

[Out]

int(((x^4 - 1)*(x^4 + 1)^(1/2))/(x^2 + 3*x^4 + x^6 + x^8 + 1), x)