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

Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0145405, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {404, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^4]/(1 - x^4),x]

[Out]

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

Rule 404

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[a/c, Subst[Int[1/(1 - 4*a*b*x^4), x], x
, x/Sqrt[a + b*x^4]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && PosQ[a*b]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 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])

Rule 203

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

Rubi steps

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

Mathematica [C]  time = 0.0909969, size = 108, normalized size = 2.04 \[ -\frac{5 x \sqrt{x^4+1} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-x^4,x^4\right )}{\left (x^4-1\right ) \left (2 x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-x^4,x^4\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-x^4,x^4\right )\right )+5 F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-x^4,x^4\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[1 + x^4]/(1 - x^4),x]

[Out]

(-5*x*Sqrt[1 + x^4]*AppellF1[1/4, -1/2, 1, 5/4, -x^4, x^4])/((-1 + x^4)*(5*AppellF1[1/4, -1/2, 1, 5/4, -x^4, x
^4] + 2*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, -x^4, x^4] + AppellF1[5/4, 1/2, 1, 9/4, -x^4, x^4])))

________________________________________________________________________________________

Maple [C]  time = 0.024, size = 365, normalized size = 6.9 \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}}}}-{\frac{{\frac{i}{2}}{\it EllipticE} \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}}}}-{\frac{{\frac{i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \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}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{\frac{i}{2}}{\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}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,-i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)^(1/2)/(-x^4+1),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)-1/2*I/(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)*EllipticE(x*(1/2*2^(
1/2)+1/2*I*2^(1/2)),I)-1/2*I/(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)*(Ellipt
icF(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),I)-EllipticE(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),I))-(-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))+1/2*I/(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)-(-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{\sqrt{x^{4} + 1}}{x^{4} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.42213, size = 173, normalized size = 3.26 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}}\right ) + \frac{1}{8} \, \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^4+1)^(1/2)/(-x^4+1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) + 1/8*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{\sqrt{x^{4} + 1}}{x^{4} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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