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

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])

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Rubi [A]  time = 0.0367356, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}} \]

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])

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Rubi in Sympy [A]  time = 2.58519, size = 46, normalized size = 0.87 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}} \right )}}{4} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+1)**(1/2)/(-x**4+1),x)

[Out]

sqrt(2)*atan(sqrt(2)*x/sqrt(x**4 + 1))/4 + sqrt(2)*atanh(sqrt(2)*x/sqrt(x**4 + 1
))/4

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Mathematica [C]  time = 0.134014, 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*Appel
lF1[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])))

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Maple [C]  time = 0.04, size = 365, normalized size = 6.9 \[ -{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,-i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{\frac{i}{2}}{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}}{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

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)*Ell
ipticF(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)*(EllipticF(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)*Ellip
ticF(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)^(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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{x^{4} + 1}}{x^{4} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.25386, size = 81, normalized size = 1.53 \[ \frac{1}{8} \, \sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}}\right ) + \log \left (\frac{\sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} + 4 \, \sqrt{x^{4} + 1} x}{x^{4} - 2 \, x^{2} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*(2*arctan(sqrt(2)*x/sqrt(x^4 + 1)) + log((sqrt(2)*(x^4 + 2*x^2 + 1)
+ 4*sqrt(x^4 + 1)*x)/(x^4 - 2*x^2 + 1)))

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

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)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{x^{4} + 1}}{x^{4} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="giac")

[Out]

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