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

Optimal. Leaf size=114 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}-\frac{5 (1-x)^{3/4} \sqrt [4]{x+1}}{12 x^2}-\frac{11 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x}-\frac{3}{8} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{3}{8} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(3*x^3) - (5*(1 - x)^(3/4)*(1 + x)^(1/4))/(12*x^2
) - (11*(1 - x)^(3/4)*(1 + x)^(1/4))/(24*x) - (3*ArcTan[(1 + x)^(1/4)/(1 - x)^(1
/4)])/8 - (3*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)])/8

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Rubi [A]  time = 0.162565, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}-\frac{5 (1-x)^{3/4} \sqrt [4]{x+1}}{12 x^2}-\frac{11 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x}-\frac{3}{8} \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac{3}{8} \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4))/(3*x^3) - (5*(1 - x)^(3/4)*(1 + x)^(1/4))/(12*x^2
) - (11*(1 - x)^(3/4)*(1 + x)^(1/4))/(24*x) - (3*ArcTan[(1 + x)^(1/4)/(1 - x)^(1
/4)])/8 - (3*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)])/8

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Rubi in Sympy [A]  time = 12.4378, size = 95, normalized size = 0.83 \[ - \frac{3 \operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{8} - \frac{3 \operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )}}{8} - \frac{11 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{24 x} - \frac{5 \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{12 x^{2}} - \frac{\left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*atan((x + 1)**(1/4)/(-x + 1)**(1/4))/8 - 3*atanh((x + 1)**(1/4)/(-x + 1)**(1/
4))/8 - 11*(-x + 1)**(3/4)*(x + 1)**(1/4)/(24*x) - 5*(-x + 1)**(3/4)*(x + 1)**(1
/4)/(12*x**2) - (-x + 1)**(3/4)*(x + 1)**(1/4)/(3*x**3)

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Mathematica [C]  time = 0.153817, size = 119, normalized size = 1.04 \[ \frac{-\frac{36 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )}{8 x F_1\left (1;\frac{1}{4},\frac{3}{4};2;\frac{1}{x},-\frac{1}{x}\right )-3 F_1\left (2;\frac{1}{4},\frac{7}{4};3;\frac{1}{x},-\frac{1}{x}\right )+F_1\left (2;\frac{5}{4},\frac{3}{4};3;\frac{1}{x},-\frac{1}{x}\right )}-\frac{8}{x^3}-\frac{10}{x^2}+11 x-\frac{3}{x}+10}{24 \sqrt [4]{1-x} (x+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10 - 8/x^3 - 10/x^2 - 3/x + 11*x - (36*x*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)] + AppellF1[2, 5/4, 3/4, 3, x^(-1), -x^(-1)]))/(24*(1 - x)^(1/4
)*(1 + x)^(3/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235137, size = 151, normalized size = 1.32 \[ \frac{18 \, x^{3} \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + 9 \, x^{3} \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 9 \, x^{3} \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - 2 \,{\left (11 \, x^{2} + 10 \, x + 8\right )}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{48 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(18*x^3*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + 9*x^3*log((x + (x +
1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - 9*x^3*log(-(x - (x + 1)^(1/4)*(-x + 1)^(
3/4) - 1)/(x - 1)) - 2*(11*x^2 + 10*x + 8)*(x + 1)^(1/4)*(-x + 1)^(3/4))/x^3

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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**4,x)

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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