3.578 \(\int \frac{1+x^3}{\left (1-x^4\right ) \sqrt [4]{1+x^4}} \, dx\)

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.41132, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 19, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.792 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{x^4+1}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: GeneratorsError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: GeneratorsError

_______________________________________________________________________________________

Mathematica [C]  time = 0.316525, size = 166, normalized size = 1.61 \[ \frac{-\log \left (1-\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )+\log \left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{2}}-\frac{2 x^4 F_1\left (1;\frac{1}{4},1;2;-x^4,x^4\right )}{\left (x^4-1\right ) \sqrt [4]{x^4+1} \left (x^4 \left (4 F_1\left (2;\frac{1}{4},2;3;-x^4,x^4\right )-F_1\left (2;\frac{5}{4},1;3;-x^4,x^4\right )\right )+8 F_1\left (1;\frac{1}{4},1;2;-x^4,x^4\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*x^4*AppellF1[1, 1/4, 1, 2, -x^4, x^4])/((-1 + x^4)*(1 + x^4)^(1/4)*(8*Appell
F1[1, 1/4, 1, 2, -x^4, x^4] + x^4*(4*AppellF1[2, 1/4, 2, 3, -x^4, x^4] - AppellF
1[2, 5/4, 1, 3, -x^4, x^4]))) + (2*ArcTan[(2^(1/4)*x)/(1 + x^4)^(1/4)] - Log[1 -
(2^(1/4)*x)/(1 + x^4)^(1/4)] + Log[1 + (2^(1/4)*x)/(1 + x^4)^(1/4)])/(4*2^(1/4)
)

_______________________________________________________________________________________

Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}+1}{-{x}^{4}+1}{\frac{1}{\sqrt [4]{{x}^{4}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)/(-x^4+1)/(x^4+1)^(1/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)), x)

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x^3 + 1)/((x^4 + 1)^(1/4)*(x^4 - 1)), x)