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

Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}} \]

[Out]

ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))]/(2*2^(3/4)) + ArcTanh[x/(2^(1/4)*(1 + x^4)^(
1/4))]/(2*2^(3/4))

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Rubi [A]  time = 0.0568344, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{2\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))]/(2*2^(3/4)) + ArcTanh[x/(2^(1/4)*(1 + x^4)^(
1/4))]/(2*2^(3/4))

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Rubi in Sympy [A]  time = 4.79372, size = 49, normalized size = 0.92 \[ \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{4} + \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*atan(2**(3/4)*x/(2*(x**4 + 1)**(1/4)))/4 + 2**(1/4)*atanh(2**(3/4)*x/(2
*(x**4 + 1)**(1/4)))/4

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Mathematica [A]  time = 0.08512, size = 70, normalized size = 1.32 \[ \frac{-\log \left (2-\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{4\ 2^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))] - Log[2 - (2^(3/4)*x)/(1 + x^4)^(1/4)] +
Log[2 + (2^(3/4)*x)/(1 + x^4)^(1/4)])/(4*2^(3/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}+2}{\frac{1}{\sqrt [4]{{x}^{4}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+1)^(1/4)/(x^4+2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 4.31309, size = 273, normalized size = 5.15 \[ \frac{1}{64} \cdot 8^{\frac{3}{4}}{\left (4 \, \arctan \left (-\frac{4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}}{2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} - 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - \sqrt{2}{\left (x^{4} + 2\right )}}\right ) + \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 4 \, \sqrt{x^{4} + 1} x^{2} + \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right ) - \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - 4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*8^(3/4)*(4*arctan(-(4*sqrt(x^4 + 1)*x^2 - sqrt(2)*(3*x^4 + 2))/(2*8^(1/4)*(
x^4 + 1)^(1/4)*x^3 - 8^(3/4)*(x^4 + 1)^(3/4)*x - sqrt(2)*(x^4 + 2))) + log(2*(2*
8^(1/4)*(x^4 + 1)^(1/4)*x^3 + 8^(3/4)*(x^4 + 1)^(3/4)*x + 4*sqrt(x^4 + 1)*x^2 +
sqrt(2)*(3*x^4 + 2))/(x^4 + 2)) - log(2*(2*8^(1/4)*(x^4 + 1)^(1/4)*x^3 + 8^(3/4)
*(x^4 + 1)^(3/4)*x - 4*sqrt(x^4 + 1)*x^2 - sqrt(2)*(3*x^4 + 2))/(x^4 + 2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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