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

Optimal. Leaf size=104 \[ \frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}} \]

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Rubi [C]  time = 0.47, antiderivative size = 139, normalized size of antiderivative = 1.34, number of steps used = 16, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6728, 406, 221, 409, 1213, 537} \begin {gather*} -\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[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 221

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 406

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 409

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 1213

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 6728

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*} \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx &=\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 ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\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 ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\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 ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}

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Mathematica [C]  time = 0.23, size = 51, normalized size = 0.49 \begin {gather*} \frac {1}{2} \left (-2 F\left (\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-\sqrt [6]{-1};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (\sqrt [6]{-1};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-(-1)^{5/6};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left ((-1)^{5/6};\left .\sin ^{-1}(x)\right |-1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*EllipticF[ArcSin[x], -1] + EllipticPi[-(-1)^(1/6), ArcSin[x], -1] + EllipticPi[(-1)^(1/6), ArcSin[x], -1]
+ EllipticPi[-(-1)^(5/6), ArcSin[x], -1] + EllipticPi[(-1)^(5/6), ArcSin[x], -1])/2

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IntegrateAlgebraic [A]  time = 0.34, size = 71, normalized size = 0.68 \begin {gather*} \frac {1}{2} (-1)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1} x \sqrt {1-x^4}}{-1+x^4}\right )+\frac {1}{2} \sqrt [4]{-1} \tanh ^{-1}\left (\frac {(-1)^{3/4} x \sqrt {1-x^4}}{-1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.62, size = 466, normalized size = 4.48 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {-x^{4} + 1} - {\left (4 \, \sqrt {-x^{4} + 1} x^{3} - \sqrt {2} {\left (x^{8} + 2 \, x^{6} - 3 \, x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} - 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} + 4 \, x^{6} - x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {-x^{4} + 1} - {\left (4 \, \sqrt {-x^{4} + 1} x^{3} + \sqrt {2} {\left (x^{8} + 2 \, x^{6} - 3 \, x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} - 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} + 4 \, x^{6} - x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} - 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} - 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 1.07, size = 114, normalized size = 1.10

method	result	size
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	\(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \sqrt {-x^{4}+1}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x +1\right ) \left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x +1\right )}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{4}+1}\, x}{\left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1\right ) \left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1\right )}\right )}{4}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{x^{8} - x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-x^4}\,\left (x^4+1\right )}{x^8-x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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