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

Optimal. Leaf size=91 \[ -\frac{(1-x)^{3/4} (x+1)^{5/4}}{2 x^2}-\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{4 x}-\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(4*x) - ((1 - x)^(3/4)*(1 + x)^(5/4))/(2*x^2) - A
rcTan[(1 + x)^(1/4)/(1 - x)^(1/4)]/4 - ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)]/4

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Rubi [A]  time = 0.09199, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{(1-x)^{3/4} (x+1)^{5/4}}{2 x^2}-\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{4 x}-\frac{1}{4} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{1}{4} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(4*x) - ((1 - x)^(3/4)*(1 + x)^(5/4))/(2*x^2) - A
rcTan[(1 + x)^(1/4)/(1 - x)^(1/4)]/4 - ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)]/4

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Rubi in Sympy [A]  time = 6.85724, size = 70, normalized size = 0.77 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{4} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{4} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{4 x} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \left (x + 1\right )^{\frac{5}{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan((x + 1)**(1/4)/(-x + 1)**(1/4))/4 - atanh((x + 1)**(1/4)/(-x + 1)**(1/4))/
4 - (-x + 1)**(3/4)*(x + 1)**(1/4)/(4*x) - (-x + 1)**(3/4)*(x + 1)**(5/4)/(2*x**
2)

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Mathematica [C]  time = 0.149638, size = 114, normalized size = 1.25 \[ \frac{-\frac{4 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )}{8 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )-3 F_1\left (2;\frac{1}{4},\frac{7}{4};3;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (2;\frac{5}{4},\frac{3}{4};3;\frac{1}{x},-\frac{1}{x}\right )}-\frac{2}{x^2}+3 x-\frac{3}{x}+2}{4 \sqrt [4]{1-x} (x+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2 - 2/x^2 - 3/x + 3*x - (4*x*AppellF1[1, 1/4, 3/4, 2, x^(-1), -x^(-1)])/(8*x*Ap
pellF1[1, 1/4, 3/4, 2, x^(-1), -x^(-1)] - 3*AppellF1[2, 1/4, 7/4, 3, x^(-1), -x^
(-1)] + AppellF1[2, 5/4, 3/4, 3, x^(-1), -x^(-1)]))/(4*(1 - x)^(1/4)*(1 + x)^(3/
4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.238865, size = 143, normalized size = 1.57 \[ \frac{2 \, x^{2} \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + x^{2} \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - x^{2} \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 2 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**3,x)

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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