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

Optimal. Leaf size=203 \[ -\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}-2 \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

[Out]

-2*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - x)^(1/
4))/(1 + x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)] -
2*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)] - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqr
t[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (
Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2]

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Rubi [A]  time = 0.202828, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}-2 \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2*ArcTan[(1 + x)^(1/4)/(1 - x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - x)^(1/
4))/(1 + x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)] -
2*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)] - Log[1 + Sqrt[1 - x]/Sqrt[1 + x] - (Sqr
t[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (
Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/Sqrt[2]

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Rubi in Sympy [A]  time = 20.788, size = 172, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{2} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{2} - 2 \operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} - 1 \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + 1 \right )} - 2 \operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*log(-sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) + sqrt(-x + 1)/sqrt(x + 1)
+ 1)/2 + sqrt(2)*log(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4) + sqrt(-x + 1)/sqrt(
x + 1) + 1)/2 - 2*atan((x + 1)**(1/4)/(-x + 1)**(1/4)) - sqrt(2)*atan(sqrt(2)*(-
x + 1)**(1/4)/(x + 1)**(1/4) - 1) - sqrt(2)*atan(sqrt(2)*(-x + 1)**(1/4)/(x + 1)
**(1/4) + 1) - 2*atanh((x + 1)**(1/4)/(-x + 1)**(1/4))

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Mathematica [C]  time = 0.207344, size = 119, normalized size = 0.59 \[ \frac{72 (x+1)^{5/4} F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{x+1}{2},x+1\right )}{5 \sqrt [4]{1-x} x \left (18 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{x+1}{2},x+1\right )+(x+1) \left (8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{x+1}{2},x+1\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{x+1}{2},x+1\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(72*(1 + x)^(5/4)*AppellF1[5/4, 1/4, 1, 9/4, (1 + x)/2, 1 + x])/(5*(1 - x)^(1/4)
*x*(18*AppellF1[5/4, 1/4, 1, 9/4, (1 + x)/2, 1 + x] + (1 + x)*(8*AppellF1[9/4, 1
/4, 2, 13/4, (1 + x)/2, 1 + x] + AppellF1[9/4, 5/4, 1, 13/4, (1 + x)/2, 1 + x]))
)

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245397, size = 443, normalized size = 2.18 \[ \frac{1}{2} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + \sqrt{2} \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - \sqrt{2} \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) + 4 \, \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - 1}\right ) + 4 \, \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + 1}\right ) - \log \left (\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x - 1}\right ) + \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1\right )}}{x - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*(2*sqrt(2)*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + sqrt(2)*lo
g((x + (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - sqrt(2)*log(-(x - (x + 1)^(1
/4)*(-x + 1)^(3/4) - 1)/(x - 1)) + 4*arctan((x - 1)/(sqrt(2)*(x - 1)*sqrt((sqrt(
2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - sqrt(x + 1)*sqrt(-x + 1) - 1)/(x - 1)) + s
qrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) + x - 1)) + 4*arctan((x - 1)/(sqrt(2)*(x - 1
)*sqrt(-(sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) - x + sqrt(x + 1)*sqrt(-x + 1) + 1
)/(x - 1)) + sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) - x + 1)) - log(2*(sqrt(2)*(x
+ 1)^(1/4)*(-x + 1)^(3/4) + x - sqrt(x + 1)*sqrt(-x + 1) - 1)/(x - 1)) + log(-2*
(sqrt(2)*(x + 1)^(1/4)*(-x + 1)^(3/4) - x + sqrt(x + 1)*sqrt(-x + 1) + 1)/(x - 1
)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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