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

Optimal. Leaf size=23 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{\sqrt{2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0321883, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1699, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{\sqrt{2}} \]

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 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]

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])

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*}

Mathematica [C]  time = 0.096379, size = 36, normalized size = 1.57 \[ \sqrt [4]{-1} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-1} x\right ),-1\right )-2 \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )\right ) \]

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])

________________________________________________________________________________________

Maple [C]  time = 0.059, size = 112, normalized size = 4.9 \begin{align*} -{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-2\,{\frac{ \left ( -1 \right ) ^{3/4}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,-i,\sqrt{-i}- \left ( -1 \right ) ^{3/4} \right ) }{\sqrt{{x}^{4}+1}}} \end{align*}

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]

-1/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*2^(1/2)+1/2*I*2^
(1/2)),I)-2*(-1)^(3/4)*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticPi((-1)^(1/4)*x,-I,(-I)^(1/2)/(-1
)^(1/4))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [B]  time = 2.33205, size = 111, normalized size = 4.83 \begin{align*} \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{align*}

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))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{x^{2} \sqrt{x^{4} + 1} - \sqrt{x^{4} + 1}}\, dx - \int \frac{1}{x^{2} \sqrt{x^{4} + 1} - \sqrt{x^{4} + 1}}\, dx \end{align*}

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

________________________________________________________________________________________

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

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)