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

Optimal. Leaf size=186 \[ -(1-x)^{3/4} \sqrt [4]{x+1}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}} \]

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4)) + 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)]/Sqrt[2] - Log[1 +
Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt[2]) +
Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt
[2])

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Rubi [A]  time = 0.175589, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -(1-x)^{3/4} \sqrt [4]{x+1}-\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-((1 - x)^(3/4)*(1 + x)^(1/4)) + 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)]/Sqrt[2] - Log[1 +
Sqrt[1 - x]/Sqrt[1 + x] - (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt[2]) +
Log[1 + Sqrt[1 - x]/Sqrt[1 + x] + (Sqrt[2]*(1 - x)^(1/4))/(1 + x)^(1/4)]/(2*Sqrt
[2])

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Rubi in Sympy [A]  time = 19.3736, size = 155, normalized size = 0.83 \[ - \left (- x + 1\right )^{\frac{3}{4}} \sqrt [4]{x + 1} - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} - 1 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x + 1)**(3/4)*(x + 1)**(1/4) - sqrt(2)*log(-sqrt(2)*(-x + 1)**(1/4)/(x + 1)**
(1/4) + sqrt(-x + 1)/sqrt(x + 1) + 1)/4 + sqrt(2)*log(sqrt(2)*(-x + 1)**(1/4)/(x
+ 1)**(1/4) + sqrt(-x + 1)/sqrt(x + 1) + 1)/4 - sqrt(2)*atan(sqrt(2)*(-x + 1)**
(1/4)/(x + 1)**(1/4) - 1)/2 - sqrt(2)*atan(sqrt(2)*(-x + 1)**(1/4)/(x + 1)**(1/4
) + 1)/2

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Mathematica [C]  time = 0.0191814, size = 43, normalized size = 0.23 \[ \sqrt [4]{x+1} \left (2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{x+1}{2}\right )-(1-x)^{3/4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1 + x)^(1/4)*(-(1 - x)^(3/4) + 2^(3/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (1 + x)
/2])

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/4)/(1-x)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244087, size = 362, normalized size = 1.95 \[ \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - 1}\right ) + \sqrt{2} \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + 1}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x - 1}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1\right )}}{x - 1}\right ) -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.92637, size = 41, normalized size = 0.22 \[ \frac{2^{\frac{3}{4}} \left (x + 1\right )^{\frac{5}{4}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{\left (x + 1\right ) e^{2 i \pi }}{2}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*(x + 1)**(5/4)*gamma(5/4)*hyper((1/4, 5/4), (9/4,), (x + 1)*exp_polar(2
*I*pi)/2)/(2*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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