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

Optimal. Leaf size=58 \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

[Out]

4*(1 - x)^(1/4) - 2*2^(1/4)*ArcTan[(1 - x)^(1/4)/2^(1/4)] - 2*2^(1/4)*ArcTanh[(1
- x)^(1/4)/2^(1/4)]

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Rubi [A]  time = 0.0561877, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(1/4)/(1 + x),x]

[Out]

4*(1 - x)^(1/4) - 2*2^(1/4)*ArcTan[(1 - x)^(1/4)/2^(1/4)] - 2*2^(1/4)*ArcTanh[(1
- x)^(1/4)/2^(1/4)]

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Rubi in Sympy [A]  time = 5.10282, size = 51, normalized size = 0.88 \[ 4 \sqrt [4]{- x + 1} - 2 \sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- x + 1}}{2} \right )} - 2 \sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- x + 1}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(-x + 1)**(1/4) - 2*2**(1/4)*atan(2**(3/4)*(-x + 1)**(1/4)/2) - 2*2**(1/4)*ata
nh(2**(3/4)*(-x + 1)**(1/4)/2)

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Mathematica [A]  time = 0.0405748, size = 85, normalized size = 1.47 \[ 4 \sqrt [4]{1-x}+\sqrt [4]{2} \log \left (2-2^{3/4} \sqrt [4]{1-x}\right )-\sqrt [4]{2} \log \left (2^{3/4} \sqrt [4]{1-x}+2\right )-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.032, size = 62, normalized size = 1.1 \[ 4\,\sqrt [4]{1-x}-2\,\sqrt [4]{2}\arctan \left ( 1/2\,\sqrt [4]{1-x}{2}^{3/4} \right ) -\sqrt [4]{2}\ln \left ({1 \left ( \sqrt [4]{1-x}+\sqrt [4]{2} \right ) \left ( \sqrt [4]{1-x}-\sqrt [4]{2} \right ) ^{-1}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(1/4)/(1+x),x)

[Out]

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

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Maxima [A]  time = 1.4985, size = 85, normalized size = 1.47 \[ -2 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}} \log \left (-\frac{2 \,{\left (2^{\frac{1}{4}} -{\left (-x + 1\right )}^{\frac{1}{4}}\right )}}{2 \cdot 2^{\frac{1}{4}} + 2 \,{\left (-x + 1\right )}^{\frac{1}{4}}}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22285, size = 105, normalized size = 1.81 \[ 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}}{\sqrt{\sqrt{2} + \sqrt{-x + 1}} +{\left (-x + 1\right )}^{\frac{1}{4}}}\right ) - 2^{\frac{1}{4}} \log \left (2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}} \log \left (-2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.12365, size = 243, normalized size = 4.19 \[ \frac{5 \sqrt [4]{-1} \sqrt [4]{x - 1} \Gamma \left (\frac{5}{4}\right )}{\Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt [4]{-2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \left (-1\right )^{\frac{3}{4}} \sqrt [4]{2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{3 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \sqrt [4]{-2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{5 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{5 \left (-1\right )^{\frac{3}{4}} \sqrt [4]{2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{7 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**(1/4)/(1+x),x)

[Out]

5*(-1)**(1/4)*(x - 1)**(1/4)*gamma(5/4)/gamma(9/4) + 5*(-2)**(1/4)*exp(-I*pi/4)*
log(-2**(3/4)*(x - 1)**(1/4)*exp_polar(I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4))
- 5*(-1)**(3/4)*2**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*(x - 1)**(1/4)*exp_polar(3*I
*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4)) - 5*(-2)**(1/4)*exp(-I*pi/4)*log(-2**(3/
4)*(x - 1)**(1/4)*exp_polar(5*I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4)) + 5*(-1)*
*(3/4)*2**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*(x - 1)**(1/4)*exp_polar(7*I*pi/4)/2
+ 1)*gamma(5/4)/(4*gamma(9/4))

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GIAC/XCAS [A]  time = 0.228981, size = 86, normalized size = 1.48 \[ -2 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}\right ) - 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}}{\rm ln}\left ({\left | -2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}} \right |}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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