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

Optimal. Leaf size=24 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]

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Rubi [A]  time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1699, 207} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 207

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

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rubi steps

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

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Mathematica [C]  time = 0.10, size = 37, normalized size = 1.54 \begin {gather*} -\sqrt [4]{-1} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2 \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1)^(1/4)*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - 2*EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1]))

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IntegrateAlgebraic [A]  time = 0.34, size = 24, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.50, size = 42, normalized size = 1.75 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.24, size = 22, normalized size = 0.92

method	result	size
elliptic	\(-\frac {\arctanh \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right ) \sqrt {2}}{2}\)	\(22\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x -\sqrt {x^{4}+1}}{\left (-1+x \right ) \left (1+x \right )}\right )}{2}\)	\(39\)
default	\(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x^4+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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