\(\int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx\) [877]

Optimal result

Integrand size = 17, antiderivative size = 67 \[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\frac {1}{4} \arctan \left (\frac {-\frac {1}{2}-x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right )-\frac {1}{4} \text {arctanh}\left (\frac {-\frac {1}{2}+x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {414} \[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=-\frac {1}{2} \arctan \left (\frac {x \left (1-x^2\right )}{\sqrt {x^4-1}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}}\right ) \]

[In]

Int[Sqrt[-1 + x^4]/(1 + x^4),x]

[Out]

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

Rule 414

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*b, 4]}, Simp[(a/(2*c*q))*A
rcTan[q*x*((a + q^2*x^2)/(a*Sqrt[a + b*x^4]))], x] + Simp[(a/(2*c*q))*ArcTanh[q*x*((a - q^2*x^2)/(a*Sqrt[a + b
*x^4]))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} \arctan \left (\frac {x \left (1-x^2\right )}{\sqrt {-1+x^4}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\left (-\frac {1}{4}+\frac {i}{4}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{x}\right ) \]

[In]

Integrate[Sqrt[-1 + x^4]/(1 + x^4),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.74 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\left (\frac {1}{4}+\frac {i}{4}\right ) \left (-\ln \left (\frac {\left (1-i\right ) \sqrt {x^{4}-1}-2 i x}{x^{2}+i}\right )+\arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^{4}-1}}{x}\right )-\ln \left (2\right )\right )\)	\(53\)
default	\(\frac {\ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}\)	\(87\)
elliptic	\(\frac {\ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}+\frac {\arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}\)	\(87\)
trager	\(\operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}+2 x}{4 x^{2} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}+1}\right )-\frac {\ln \left (\frac {4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x -\sqrt {x^{4}-1}}{4 x^{2} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right )}{2}-\ln \left (\frac {4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x -\sqrt {x^{4}-1}}{4 x^{2} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )\)	\(178\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\frac {1}{2} \, \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 1}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\int { \frac {\sqrt {x^{4} - 1}}{x^{4} + 1} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^4 - 1)/(x^4 + 1), x)

Giac [F]

\[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\int { \frac {\sqrt {x^{4} - 1}}{x^{4} + 1} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^4 - 1)/(x^4 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx=\int \frac {\sqrt {x^4-1}}{x^4+1} \,d x \]

[In]

int((x^4 - 1)^(1/2)/(x^4 + 1),x)

[Out]

int((x^4 - 1)^(1/2)/(x^4 + 1), x)