\(\int \frac {\sqrt {1-x^4} (1+x^4)}{1-x^4+x^8} \, dx\) [1498]

Optimal result

Integrand size = 29, antiderivative size = 104 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}}-\frac {\text {arctanh}\left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.34, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6860, 415, 227, 418, 1227, 551} \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{2} \left (1+i \sqrt {3}\right ) \operatorname {EllipticF}(\arcsin (x),-1)-\frac {1}{2} \left (1-i \sqrt {3}\right ) \operatorname {EllipticF}(\arcsin (x),-1)+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-i-\sqrt {3}\right ),\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (i-\sqrt {3}\right ),\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}},\arcsin (x),-1\right )+\frac {1}{2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}},\arcsin (x),-1\right ) \]

[In]

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

[Out]

-1/2*((1 - I*Sqrt[3])*EllipticF[ArcSin[x], -1]) - ((1 + I*Sqrt[3])*EllipticF[ArcSin[x], -1])/2 + EllipticPi[(-
I - Sqrt[3])/2, ArcSin[x], -1]/2 + EllipticPi[(I - Sqrt[3])/2, ArcSin[x], -1]/2 + EllipticPi[1/Sqrt[(1 - I*Sqr
t[3])/2], ArcSin[x], -1]/2 + EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[x], -1]/2

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 415

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=-\frac {1}{2} (-1)^{3/4} \left (\arctan \left (\frac {(1+i) x}{\sqrt {2-2 x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-1} x}{\sqrt {1-x^4}}\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.53

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\left (-1+i\right ) \arctan \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )+\left (-1-i\right ) \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )\right )}{4}\)	\(55\)
default	\(\frac {\left (\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\)	\(114\)
elliptic	\(\frac {\left (\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\)	\(114\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{4}+1}\, x}{\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1\right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1\right )}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \sqrt {-x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +1\right ) \left (\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x +1\right )}\right )}{4}\)	\(217\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.66 \[ \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x^{8} + \left (2 i - 2\right ) \, x^{6} - \left (3 i + 3\right ) \, x^{4} - \left (2 i - 2\right ) \, x^{2} + i + 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x^{8} - \left (2 i + 2\right ) \, x^{6} + \left (3 i - 3\right ) \, x^{4} + \left (2 i + 2\right ) \, x^{2} - i + 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x^{8} + \left (2 i + 2\right ) \, x^{6} - \left (3 i - 3\right ) \, x^{4} - \left (2 i + 2\right ) \, x^{2} + i - 1\right )} - 4 \, {\left (x^{5} - i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x^{8} - \left (2 i - 2\right ) \, x^{6} + \left (3 i + 3\right ) \, x^{4} + \left (2 i - 2\right ) \, x^{2} - i - 1\right )} - 4 \, {\left (x^{5} + i \, x^{3} - x\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\right ) \]

[In]

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

[Out]

(1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*((I + 1)*x^8 + (2*I - 2)*x^6 - (3*I + 3)*x^4 - (2*I - 2)*x^2 + I + 1) - 4
*(x^5 + I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1)) - (1/16*I - 1/16)*sqrt(2)*log((sqrt(2)*(-(I - 1)*x^8 - (2*
I + 2)*x^6 + (3*I - 3)*x^4 + (2*I + 2)*x^2 - I + 1) - 4*(x^5 - I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1)) + (
1/16*I - 1/16)*sqrt(2)*log((sqrt(2)*((I - 1)*x^8 + (2*I + 2)*x^6 - (3*I - 3)*x^4 - (2*I + 2)*x^2 + I - 1) - 4*
(x^5 - I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1)) - (1/16*I + 1/16)*sqrt(2)*log((sqrt(2)*(-(I + 1)*x^8 - (2*I
- 2)*x^6 + (3*I + 3)*x^4 + (2*I - 2)*x^2 - I - 1) - 4*(x^5 + I*x^3 - x)*sqrt(-x^4 + 1))/(x^8 - x^4 + 1))

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))*(x**4 + 1)/(x**8 - x**4 + 1), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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