3.884 \(\int \frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{x}-\tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-(((1 - x)^(3/4)*(1 + x)^(1/4))/x) - ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] - ArcTa
nh[(1 + x)^(1/4)/(1 - x)^(1/4)]

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Rubi [A]  time = 0.0647105, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{x}-\tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(((1 - x)^(3/4)*(1 + x)^(1/4))/x) - ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] - ArcTa
nh[(1 + x)^(1/4)/(1 - x)^(1/4)]

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Rubi in Sympy [A]  time = 5.27498, size = 46, normalized size = 0.74 \[ - \operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} - \operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.24263, size = 106, normalized size = 1.71 \[ \frac{-\frac{4 x^2 F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )}{8 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )-3 F_1\left (2;\frac{1}{4},\frac{7}{4};3;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (2;\frac{5}{4},\frac{3}{4};3;\frac{1}{x},-\frac{1}{x}\right )}+x^2-1}{\sqrt [4]{1-x} x (x+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-1 + x^2 - (4*x^2*AppellF1[1, 1/4, 3/4, 2, x^(-1), -x^(-1)])/(8*x*AppellF1[1, 1
/4, 3/4, 2, x^(-1), -x^(-1)] - 3*AppellF1[2, 1/4, 7/4, 3, x^(-1), -x^(-1)] + App
ellF1[2, 5/4, 3/4, 3, x^(-1), -x^(-1)]))/((1 - x)^(1/4)*x*(1 + x)^(3/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/4)/(1-x)^(1/4)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245045, size = 128, normalized size = 2.06 \[ \frac{2 \, x \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + x \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - x \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 2 \,{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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