3.2.18 \(\int \frac {-1+x^4}{(1+x^4) \sqrt {1+x^2+x^4}} \, dx\)

Optimal. Leaf size=17 \[ -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2112, 206} \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + x^2 + x^4]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2112

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps

\begin {align*} \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt {1+x^2+x^4}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.23, size = 120, normalized size = 7.06 \begin {gather*} \frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-\Pi \left ((-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{\sqrt {x^4+x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]
- EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - EllipticPi[(-1)^(5/6), I*ArcSinh[(-1)^(5/6)*x
], (-1)^(2/3)]))/Sqrt[1 + x^2 + x^4]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.24, size = 17, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^4)/((1 + x^4)*Sqrt[1 + x^2 + x^4]),x]

[Out]

-ArcTanh[x/Sqrt[1 + x^2 + x^4]]

________________________________________________________________________________________

fricas [B]  time = 0.48, size = 34, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + 1} x + 1}{x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.29, size = 18, normalized size = 1.06

method	result	size
elliptic	\(-\arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\)	\(18\)
trager	\(\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+x^{2}+1}\, x -2 x^{2}-1}{x^{4}+1}\right )}{2}\)	\(38\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {i \sqrt {3}-1}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{4}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x^4-1}{\left (x^4+1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{4} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________