3.315 \(\int \frac{1}{\left (1+x^4\right ) \sqrt [4]{2+x^4}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+1\right )}{2 \sqrt{2}}-\frac{\log \left (-\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+\frac{x^2}{\sqrt{x^4+2}}+1\right )}{4 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+\frac{x^2}{\sqrt{x^4+2}}+1\right )}{4 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.137354, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+1\right )}{2 \sqrt{2}}-\frac{\log \left (-\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+\frac{x^2}{\sqrt{x^4+2}}+1\right )}{4 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{2} x}{\sqrt [4]{x^4+2}}+\frac{x^2}{\sqrt{x^4+2}}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.5697, size = 124, normalized size = 0.88 \[ - \frac{\sqrt{2} \log{\left (\frac{x^{2}}{\sqrt{x^{4} + 2}} - \frac{\sqrt{2} x}{\sqrt [4]{x^{4} + 2}} + 1 \right )}}{8} + \frac{\sqrt{2} \log{\left (\frac{x^{2}}{\sqrt{x^{4} + 2}} + \frac{\sqrt{2} x}{\sqrt [4]{x^{4} + 2}} + 1 \right )}}{8} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt [4]{x^{4} + 2}} - 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt [4]{x^{4} + 2}} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.113781, size = 132, normalized size = 0.94 \[ \frac{-2 \tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{2 x^4+1}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{2 x^4+1}}+1\right )-\log \left (-\frac{\sqrt{2} x}{\sqrt [4]{2 x^4+1}}+\frac{x^2}{\sqrt{2 x^4+1}}+1\right )+\log \left (\frac{\sqrt{2} x}{\sqrt [4]{2 x^4+1}}+\frac{x^2}{\sqrt{2 x^4+1}}+1\right )}{4 \sqrt{2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}+1}{\frac{1}{\sqrt [4]{{x}^{4}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+1)/(x^4+2)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 3.52353, size = 524, normalized size = 3.72 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (-\frac{{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + \sqrt{2} \sqrt{x^{4} + 2} x^{2} +{\left (x^{4} + 2\right )}^{\frac{3}{4}} x}{{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} - \sqrt{2}{\left (x^{4} + 1\right )} \sqrt{\frac{x^{4} + \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + 2 \, \sqrt{x^{4} + 2} x^{2} + \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{3}{4}} x + 1}{x^{4} + 1}} -{\left (x^{4} + 2\right )}^{\frac{3}{4}} x - \sqrt{2}}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} - \sqrt{2} \sqrt{x^{4} + 2} x^{2} +{\left (x^{4} + 2\right )}^{\frac{3}{4}} x}{{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + \sqrt{2}{\left (x^{4} + 1\right )} \sqrt{\frac{x^{4} - \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + 2 \, \sqrt{x^{4} + 2} x^{2} - \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{3}{4}} x + 1}{x^{4} + 1}} -{\left (x^{4} + 2\right )}^{\frac{3}{4}} x + \sqrt{2}}\right ) + \frac{1}{16} \, \sqrt{2} \log \left (\frac{2 \,{\left (x^{4} + \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + 2 \, \sqrt{x^{4} + 2} x^{2} + \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{3}{4}} x + 1\right )}}{x^{4} + 1}\right ) - \frac{1}{16} \, \sqrt{2} \log \left (\frac{2 \,{\left (x^{4} - \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{1}{4}} x^{3} + 2 \, \sqrt{x^{4} + 2} x^{2} - \sqrt{2}{\left (x^{4} + 2\right )}^{\frac{3}{4}} x + 1\right )}}{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*arctan(-((x^4 + 2)^(1/4)*x^3 + sqrt(2)*sqrt(x^4 + 2)*x^2 + (x^4 + 2)
^(3/4)*x)/((x^4 + 2)^(1/4)*x^3 - sqrt(2)*(x^4 + 1)*sqrt((x^4 + sqrt(2)*(x^4 + 2)
^(1/4)*x^3 + 2*sqrt(x^4 + 2)*x^2 + sqrt(2)*(x^4 + 2)^(3/4)*x + 1)/(x^4 + 1)) - (
x^4 + 2)^(3/4)*x - sqrt(2))) + 1/4*sqrt(2)*arctan(((x^4 + 2)^(1/4)*x^3 - sqrt(2)
*sqrt(x^4 + 2)*x^2 + (x^4 + 2)^(3/4)*x)/((x^4 + 2)^(1/4)*x^3 + sqrt(2)*(x^4 + 1)
*sqrt((x^4 - sqrt(2)*(x^4 + 2)^(1/4)*x^3 + 2*sqrt(x^4 + 2)*x^2 - sqrt(2)*(x^4 +
2)^(3/4)*x + 1)/(x^4 + 1)) - (x^4 + 2)^(3/4)*x + sqrt(2))) + 1/16*sqrt(2)*log(2*
(x^4 + sqrt(2)*(x^4 + 2)^(1/4)*x^3 + 2*sqrt(x^4 + 2)*x^2 + sqrt(2)*(x^4 + 2)^(3/
4)*x + 1)/(x^4 + 1)) - 1/16*sqrt(2)*log(2*(x^4 - sqrt(2)*(x^4 + 2)^(1/4)*x^3 + 2
*sqrt(x^4 + 2)*x^2 - sqrt(2)*(x^4 + 2)^(3/4)*x + 1)/(x^4 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x^{4} + 1\right ) \sqrt [4]{x^{4} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 2\right )}^{\frac{1}{4}}{\left (x^{4} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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