3.1219 \(\int \frac{1}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2} \sqrt [4]{b}}-\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} \sqrt [4]{b}}+\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} \sqrt [4]{b}} \]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) + ArcTan[
1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[1 + (Sqrt[b
]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(4*Sqrt[2]*b^(1/
4)) + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1
/4)]/(4*Sqrt[2]*b^(1/4))

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Rubi [A]  time = 0.185019, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2} \sqrt [4]{b}}-\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} \sqrt [4]{b}}+\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(-1/4),x]

[Out]

-ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) + ArcTan[
1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[1 + (Sqrt[b
]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]/(4*Sqrt[2]*b^(1/
4)) + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1
/4)]/(4*Sqrt[2]*b^(1/4))

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Rubi in Sympy [A]  time = 28.7986, size = 185, normalized size = 0.89 \[ - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8 \sqrt [4]{b}} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8 \sqrt [4]{b}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{4 \sqrt [4]{b}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{4 \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(a - b*x
**4) + 1)/(8*b**(1/4)) + sqrt(2)*log(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sq
rt(b)*x**2/sqrt(a - b*x**4) + 1)/(8*b**(1/4)) + sqrt(2)*atan(sqrt(2)*b**(1/4)*x/
(a - b*x**4)**(1/4) - 1)/(4*b**(1/4)) + sqrt(2)*atan(sqrt(2)*b**(1/4)*x/(a - b*x
**4)**(1/4) + 1)/(4*b**(1/4))

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Mathematica [A]  time = 0.0346532, size = 173, normalized size = 0.83 \[ \frac{-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )+\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(-1/4),x]

[Out]

(-2*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*b^
(1/4)*x)/(a - b*x^4)^(1/4)] - Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b
^(1/4)*x)/(a - b*x^4)^(1/4)] + Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*
b^(1/4)*x)/(a - b*x^4)^(1/4)])/(4*Sqrt[2]*b^(1/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^4+a)^(1/4),x)

[Out]

int(1/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238695, size = 190, normalized size = 0.91 \[ -\left (-\frac{1}{b}\right )^{\frac{1}{4}} \arctan \left (\frac{b x \left (-\frac{1}{b}\right )^{\frac{3}{4}}}{x \sqrt{-\frac{b x^{2} \sqrt{-\frac{1}{b}} - \sqrt{-b x^{4} + a}}{x^{2}}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \left (-\frac{1}{b}\right )^{\frac{1}{4}} \log \left (\frac{b x \left (-\frac{1}{b}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{4} \, \left (-\frac{1}{b}\right )^{\frac{1}{4}} \log \left (-\frac{b x \left (-\frac{1}{b}\right )^{\frac{3}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.70816, size = 37, normalized size = 0.18 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/4), (5/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(1/4)*g
amma(5/4))

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GIAC/XCAS [A]  time = 0.232224, size = 238, normalized size = 1.14 \[ -\frac{\sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{4 \, b^{\frac{1}{4}}} - \frac{\sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{4 \, b^{\frac{1}{4}}} + \frac{\sqrt{2}{\rm ln}\left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{8 \, b^{\frac{1}{4}}} - \frac{\sqrt{2}{\rm ln}\left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{8 \, b^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4)/x)/b^(1/
4))/b^(1/4) - 1/4*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-b*x^4 + a)^
(1/4)/x)/b^(1/4))/b^(1/4) + 1/8*sqrt(2)*ln(sqrt(b) + sqrt(2)*(-b*x^4 + a)^(1/4)*
b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(1/4) - 1/8*sqrt(2)*ln(sqrt(b) - sqrt(2)*(-b
*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(1/4)