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

Optimal. Leaf size=21 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {2 x^4-x^2-1}}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2112, 204} \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {2 x^4-x^2-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[x/Sqrt[-1 - x^2 + 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+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {x}{\sqrt {-1-x^2+2 x^4}}\right )\\ &=-\tan ^{-1}\left (\frac {x}{\sqrt {-1-x^2+2 x^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.27, size = 99, normalized size = 4.71 \begin {gather*} -\frac {i \sqrt {-2 x^4+x^2+1} \left (F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )-\Pi \left (-\frac {1}{\sqrt {2}};i \sinh ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )-\Pi \left (\frac {1}{\sqrt {2}};i \sinh ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )\right )}{\sqrt {4 x^4-2 x^2-2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 21, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {-1-x^2+2 x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTan[x/Sqrt[-1 - x^2 + 2*x^4]]

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fricas [A]  time = 0.50, size = 34, normalized size = 1.62 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {2 \, x^{4} - x^{2} - 1} x}{2 \, x^{4} - 2 \, x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(2*sqrt(2*x^4 - x^2 - 1)*x/(2*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 {2 \, x^{4} + 1}{\sqrt {2 \, x^{4} - x^{2} - 1} {\left (2 \, x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.32, size = 20, normalized size = 0.95

method	result	size
elliptic	\(\arctan \left (\frac {\sqrt {2 x^{4}-x^{2}-1}}{x}\right )\)	\(20\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {2 x^{4}-x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{4}-1}\right )}{2}\)	\(69\)
default	\(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x \sqrt {2}, \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {2 x^{4}-x^{2}-1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (-9 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 x^{2}-4\right )}{14 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {2 x^{4}-x^{2}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {2 x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (\sqrt {-2}\, x , -\underline {\hspace {1.25 ex}}\alpha ^{2}, \frac {\sqrt {-2}}{2}\right )}{\sqrt {2 x^{4}-x^{2}-1}}\right )\right )}{4}\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(1/x*(2*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 {2 \, x^{4} + 1}{\sqrt {2 \, x^{4} - x^{2} - 1} {\left (2 \, x^{4} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {2\,x^4+1}{\left (2\,x^4-1\right )\,\sqrt {2\,x^4-x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*x^4 + 1)/((2*x^4 - 1)*(2*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 {2 x^{4} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (2 x^{2} + 1\right )} \left (2 x^{4} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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