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

Optimal. Leaf size=19 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {x^4-x^2-1}}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 19, 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, 204} \begin {gather*} -\tan ^{-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]

-ArcTan[x/Sqrt[-1 - x^2 + x^4]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 )\\ &=-\tan ^{-1}\left (\frac {x}{\sqrt {-1-x^2+x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 4.99, size = 1511, normalized size = 79.53

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]

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

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IntegrateAlgebraic [A]  time = 0.25, size = 19, normalized size = 1.00 \begin {gather*} -\tan ^{-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]

-ArcTan[x/Sqrt[-1 - x^2 + x^4]]

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fricas [A]  time = 0.47, size = 30, normalized size = 1.58 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} x}{x^{4} - 2 \, x^{2} - 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*arctan(2*sqrt(x^4 - x^2 - 1)*x/(x^4 - 2*x^2 - 1))

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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)

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maple [A]  time = 0.41, size = 18, normalized size = 0.95

method	result	size
elliptic	\(\arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}}{x}\right )\)	\(18\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{4}-x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(73\)
default	\(\frac {2 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{\sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {\sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}-\frac {\sqrt {1+\frac {x^{2}}{2}+\frac {\sqrt {5}\, x^{2}}{2}}\, \sqrt {1-\frac {\sqrt {5}\, x^{2}}{2}+\frac {x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , -\frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\)	\(273\)

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]

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

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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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \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^4 - x^2 - 1)^(1/2)),x)

[Out]

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

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

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