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

Optimal. Leaf size=216 \[ -\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt{2} \sqrt{e}} \]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e])) +
ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e]) - L
og[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1 - x^2] - (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)]/(2
*Sqrt[2]*Sqrt[e]) + Log[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1 - x^2] + (Sqrt[2]*Sqrt[e*x]
)/(1 - x^2)^(1/4)]/(2*Sqrt[2]*Sqrt[e])

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Rubi [A]  time = 0.298455, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{2 \sqrt{2} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{\sqrt{2} \sqrt{e}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt{2} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e])) +
ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e]) - L
og[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1 - x^2] - (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)]/(2
*Sqrt[2]*Sqrt[e]) + Log[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1 - x^2] + (Sqrt[2]*Sqrt[e*x]
)/(1 - x^2)^(1/4)]/(2*Sqrt[2]*Sqrt[e])

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Rubi in Sympy [A]  time = 31.748, size = 185, normalized size = 0.86 \[ - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{4 \sqrt{e}} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{4 \sqrt{e}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{2 \sqrt{e}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{2 \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(1/2)/(1+x)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.0169169, size = 23, normalized size = 0.11 \[ \frac{2 x \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};x^2\right )}{\sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Hypergeometric2F1[1/4, 1/4, 5/4, x^2])/Sqrt[e*x]

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt{ex}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(1/2)/(1+x)^(1/4),x)

[Out]

int(1/(1-x)^(1/4)/(e*x)^(1/2)/(1+x)^(1/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246893, size = 626, normalized size = 2.9 \[ \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e x^{2} - e\right )} \frac{1}{e^{2}}^{\frac{1}{4}}}{2 \, \sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \sqrt{2}{\left (e x^{2} - e\right )} \frac{1}{e^{2}}^{\frac{1}{4}} + 2 \,{\left (x^{2} - 1\right )} \sqrt{\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} - e \sqrt{x + 1} x \sqrt{-x + 1} +{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}}}\right ) + \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e x^{2} - e\right )} \frac{1}{e^{2}}^{\frac{1}{4}}}{2 \, \sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{2}{\left (e x^{2} - e\right )} \frac{1}{e^{2}}^{\frac{1}{4}} + 2 \,{\left (x^{2} - 1\right )} \sqrt{-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} + e \sqrt{x + 1} x \sqrt{-x + 1} -{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}}}\right ) + \frac{1}{4} \, \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \log \left (-\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} + e \sqrt{x + 1} x \sqrt{-x + 1} -{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}\right ) - \frac{1}{4} \, \sqrt{2} \frac{1}{e^{2}}^{\frac{1}{4}} \log \left (\frac{\sqrt{2} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} \frac{1}{e^{2}}^{\frac{1}{4}} - e \sqrt{x + 1} x \sqrt{-x + 1} +{\left (e^{2} x^{2} - e^{2}\right )} \sqrt{\frac{1}{e^{2}}}}{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*(e^(-2))^(1/4)*arctan(sqrt(2)*(e*x^2 - e)*(e^(-2))^(1/4)/(2*sqrt(e*x)*(x
+ 1)^(3/4)*(-x + 1)^(3/4) + sqrt(2)*(e*x^2 - e)*(e^(-2))^(1/4) + 2*(x^2 - 1)*sq
rt((sqrt(2)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4)*(e^(-2))^(1/4) - e*sqrt(x +
1)*x*sqrt(-x + 1) + (e^2*x^2 - e^2)*sqrt(e^(-2)))/(x^2 - 1)))) + sqrt(2)*(e^(-2
))^(1/4)*arctan(sqrt(2)*(e*x^2 - e)*(e^(-2))^(1/4)/(2*sqrt(e*x)*(x + 1)^(3/4)*(-
x + 1)^(3/4) - sqrt(2)*(e*x^2 - e)*(e^(-2))^(1/4) + 2*(x^2 - 1)*sqrt(-(sqrt(2)*s
qrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4)*(e^(-2))^(1/4) + e*sqrt(x + 1)*x*sqrt(-x
+ 1) - (e^2*x^2 - e^2)*sqrt(e^(-2)))/(x^2 - 1)))) + 1/4*sqrt(2)*(e^(-2))^(1/4)*
log(-(sqrt(2)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4)*(e^(-2))^(1/4) + e*sqrt(x
+ 1)*x*sqrt(-x + 1) - (e^2*x^2 - e^2)*sqrt(e^(-2)))/(x^2 - 1)) - 1/4*sqrt(2)*(e
^(-2))^(1/4)*log((sqrt(2)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4)*(e^(-2))^(1/4
) - e*sqrt(x + 1)*x*sqrt(-x + 1) + (e^2*x^2 - e^2)*sqrt(e^(-2)))/(x^2 - 1))

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Sympy [A]  time = 77.7521, size = 90, normalized size = 0.42 \[ - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{3}{8}, \frac{7}{8} & \frac{1}{2}, \frac{3}{4}, 1, 1 \\0, \frac{3}{8}, \frac{1}{2}, \frac{7}{8}, 1, 0 & \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \sqrt{e} \Gamma \left (\frac{1}{4}\right )} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{8}, \frac{1}{4}, \frac{3}{8}, \frac{3}{4}, 1 & \\- \frac{1}{8}, \frac{3}{8} & - \frac{1}{4}, 0, \frac{1}{4}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \sqrt{e} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x)**(1/4)/(e*x)**(1/2)/(1+x)**(1/4),x)

[Out]

-I*meijerg(((3/8, 7/8), (1/2, 3/4, 1, 1)), ((0, 3/8, 1/2, 7/8, 1, 0), ()), exp_p
olar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*sqrt(e)*gamma(1/4)) - meijerg(((-1/4, -1/8
, 1/4, 3/8, 3/4, 1), ()), ((-1/8, 3/8), (-1/4, 0, 1/4, 0)), x**(-2))/(4*pi*sqrt(
e)*gamma(1/4))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out