3.317 \(\int \frac{\left (1+x^4\right )^{3/4}}{\left (2+x^4\right )^2} \, dx\)

Optimal. Leaf size=74 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]

[Out]

(x*(1 + x^4)^(3/4))/(8*(2 + x^4)) + (3*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))])/(16*
2^(3/4)) + (3*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))])/(16*2^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0641195, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}}+\frac{3 \tanh ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )}{16\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)^(3/4)/(2 + x^4)^2,x]

[Out]

(x*(1 + x^4)^(3/4))/(8*(2 + x^4)) + (3*ArcTan[x/(2^(1/4)*(1 + x^4)^(1/4))])/(16*
2^(3/4)) + (3*ArcTanh[x/(2^(1/4)*(1 + x^4)^(1/4))])/(16*2^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.68551, size = 70, normalized size = 0.95 \[ \frac{x \left (x^{4} + 1\right )^{\frac{3}{4}}}{8 \left (x^{4} + 2\right )} + \frac{3 \sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{32} + \frac{3 \sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} x}{2 \sqrt [4]{x^{4} + 1}} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+1)**(3/4)/(x**4+2)**2,x)

[Out]

x*(x**4 + 1)**(3/4)/(8*(x**4 + 2)) + 3*2**(1/4)*atan(2**(3/4)*x/(2*(x**4 + 1)**(
1/4)))/32 + 3*2**(1/4)*atanh(2**(3/4)*x/(2*(x**4 + 1)**(1/4)))/32

_______________________________________________________________________________________

Mathematica [A]  time = 0.160765, size = 92, normalized size = 1.24 \[ \frac{\left (x^4+1\right )^{3/4} x}{8 \left (x^4+2\right )}+\frac{3 \left (-\log \left (2-\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{2^{3/4} x}{\sqrt [4]{x^4+1}}+2\right )+2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )\right )}{32\ 2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)^(3/4)/(2 + x^4)^2,x]

[Out]

(x*(1 + x^4)^(3/4))/(8*(2 + x^4)) + (3*(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)]))/(3
2*2^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ({x}^{4}+2 \right ) ^{2}} \left ({x}^{4}+1 \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)^(3/4)/(x^4+2)^2,x)

[Out]

int((x^4+1)^(3/4)/(x^4+2)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 1\right )}^{\frac{3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="maxima")

[Out]

integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2, x)

_______________________________________________________________________________________

Fricas [A]  time = 4.28261, size = 323, normalized size = 4.36 \[ \frac{8^{\frac{3}{4}}{\left (8 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 12 \,{\left (x^{4} + 2\right )} \arctan \left (-\frac{4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}}{2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} - 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - \sqrt{2}{\left (x^{4} + 2\right )}}\right ) + 3 \,{\left (x^{4} + 2\right )} \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x + 4 \, \sqrt{x^{4} + 1} x^{2} + \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right ) - 3 \,{\left (x^{4} + 2\right )} \log \left (\frac{2 \,{\left (2 \cdot 8^{\frac{1}{4}}{\left (x^{4} + 1\right )}^{\frac{1}{4}} x^{3} + 8^{\frac{3}{4}}{\left (x^{4} + 1\right )}^{\frac{3}{4}} x - 4 \, \sqrt{x^{4} + 1} x^{2} - \sqrt{2}{\left (3 \, x^{4} + 2\right )}\right )}}{x^{4} + 2}\right )\right )}}{512 \,{\left (x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="fricas")

[Out]

1/512*8^(3/4)*(8*8^(1/4)*(x^4 + 1)^(3/4)*x + 12*(x^4 + 2)*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))) + 3*(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)) - 3*(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)))/(x^4 + 2)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{4} + 1\right )^{\frac{3}{4}}}{\left (x^{4} + 2\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)**(3/4)/(x**4+2)**2,x)

[Out]

Integral((x**4 + 1)**(3/4)/(x**4 + 2)**2, x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 1\right )}^{\frac{3}{4}}}{{\left (x^{4} + 2\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2,x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(3/4)/(x^4 + 2)^2, x)