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

Optimal. Leaf size=15 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2-1}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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 = 3.51, size = 1547, normalized size = 103.13

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((2*I)*Sqrt[1 - x^2 - x^4]*EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]*x], -3/2 - Sqrt[5]/2])/Sqrt[-1 + Sqrt[5]
] - ((2*I)*Sqrt[1 - x^2 - x^4]*EllipticPi[(1 + Sqrt[5])/2, I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/
2])/Sqrt[-1 + Sqrt[5]] - ((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x)^2*Sqrt[(I*
Sqrt[2*(1 + Sqrt[5])] + 2*x)/((1 + 2*I)*Sqrt[2] - Sqrt[10] + 2*Sqrt[-1 + Sqrt[5]]*x - (2*I)*Sqrt[1 + Sqrt[5]]*
x)]*Sqrt[(I*Sqrt[2*(1 + Sqrt[5])] - 2*x)/(Sqrt[2]*((-1 + 2*I) + Sqrt[5]) - 2*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 +
Sqrt[5]])*x)]*Sqrt[(Sqrt[2]*((-1 - 2*I) + Sqrt[5]) + 2*(Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*x)/(Sqrt[2]*
((-1 + 2*I) + Sqrt[5]) - 2*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*x)]*((2 + Sqrt[2*(-1 + Sqrt[5])])*Ellipt
icF[ArcSin[Sqrt[((Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] + 2*x))/((Sqrt[-1 + Sqrt[5
]] + I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x))]], -3/5 - (4*I)/5] - 2*Sqrt[2*(-1 + Sqrt[5])]*Ellipt
icPi[((-2 + Sqrt[2*(-1 + Sqrt[5])])*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]]))/((2 + Sqrt[2*(-1 + Sqrt[5])])*
(Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])), ArcSin[Sqrt[((Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(
-1 + Sqrt[5])] + 2*x))/((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x))]], -3/5 - (
4*I)/5]))/((-2 + Sqrt[2*(-1 + Sqrt[5])])*(2 + Sqrt[2*(-1 + Sqrt[5])])*(Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]
])) + ((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x)^2*Sqrt[(I*Sqrt[2*(1 + Sqrt[5]
)] + 2*x)/((1 + 2*I)*Sqrt[2] - Sqrt[10] + 2*Sqrt[-1 + Sqrt[5]]*x - (2*I)*Sqrt[1 + Sqrt[5]]*x)]*Sqrt[(I*Sqrt[2*
(1 + Sqrt[5])] - 2*x)/(Sqrt[2]*((-1 + 2*I) + Sqrt[5]) - 2*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*x)]*Sqrt[
(Sqrt[2]*((-1 - 2*I) + Sqrt[5]) + 2*(Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*x)/(Sqrt[2]*((-1 + 2*I) + Sqrt[
5]) - 2*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*x)]*((-2 + Sqrt[2*(-1 + Sqrt[5])])*EllipticF[ArcSin[Sqrt[((
Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] + 2*x))/((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sq
rt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x))]], -3/5 - (4*I)/5] - 2*Sqrt[2*(-1 + Sqrt[5])]*EllipticPi[((2 + Sqrt[2*
(-1 + Sqrt[5])])*(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]]))/((-2 + Sqrt[2*(-1 + Sqrt[5])])*(Sqrt[-1 + Sqrt[5]
] - I*Sqrt[1 + Sqrt[5]])), ArcSin[Sqrt[((Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] + 2
*x))/((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*(Sqrt[2*(-1 + Sqrt[5])] - 2*x))]], -3/5 - (4*I)/5]))/((-2 + S
qrt[2*(-1 + Sqrt[5])])*(2 + Sqrt[2*(-1 + Sqrt[5])])*(Sqrt[-1 + Sqrt[5]] - I*Sqrt[1 + Sqrt[5]])))/(Sqrt[2]*Sqrt
[-1 + x^2 + x^4])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.23, size = 15, 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 = 0.49, size = 34, normalized size = 2.27 \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 = 0.32, size = 16, normalized size = 1.07

method	result	size
elliptic	\(\arctanh \left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\)	\(16\)
trager	\(\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+x^{2}-1}\, x +2 x^{2}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(46\)
default	\(-\frac {2 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {2-2 \sqrt {5}}}{2}, \frac {i}{2}+\frac {i \sqrt {5}}{2}\right )}{\sqrt {2-2 \sqrt {5}}\, \sqrt {x^{4}+x^{2}-1}}+\frac {\sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}+\frac {\sqrt {1-\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {x^{2}}{2}-\frac {\sqrt {5}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}}}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}+x^{2}-1}}\)	\(265\)

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

________________________________________________________________________________________

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 = -0.07 \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 {x^{4}}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, dx - \int \frac {1}{x^{4} \sqrt {x^{4} + x^{2} - 1} - \sqrt {x^{4} + x^{2} - 1}}\, 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**4/(x**4*sqrt(x**4 + x**2 - 1) - sqrt(x**4 + x**2 - 1)), x) - Integral(1/(x**4*sqrt(x**4 + x**2 -
1) - sqrt(x**4 + x**2 - 1)), x)

________________________________________________________________________________________