3.1095 \(\int \frac{1}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}} \]

[Out]

ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(1/4)) + ArcTanh[(b^(1/4)*x)/(a + b*x
^4)^(1/4)]/(2*b^(1/4))

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Rubi [A]  time = 0.0334872, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(-1/4),x]

[Out]

ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(1/4)) + ArcTanh[(b^(1/4)*x)/(a + b*x
^4)^(1/4)]/(2*b^(1/4))

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Rubi in Sympy [A]  time = 4.0425, size = 49, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 \sqrt [4]{b}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*b**(1/4)) + atanh(b**(1/4)*x/(a + b*x**4
)**(1/4))/(2*b**(1/4))

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Mathematica [A]  time = 0.011137, size = 76, normalized size = 1.33 \[ \frac{-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4 \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(-1/4),x]

[Out]

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

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Maple [F]  time = 0.051, 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 = 0.289285, size = 136, normalized size = 2.39 \[ \frac{\arctan \left (\frac{b^{\frac{1}{4}} x}{x \sqrt{\frac{\sqrt{b} x^{2} + \sqrt{b x^{4} + a}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right )}{b^{\frac{1}{4}}} + \frac{\log \left (\frac{b^{\frac{1}{4}} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}{4 \, b^{\frac{1}{4}}} - \frac{\log \left (-\frac{b^{\frac{1}{4}} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}{4 \, b^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.53411, size = 36, normalized size = 0.63 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{5}{4}\right )} \]

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(I*pi)/a)/(4*a**(1/4)*gam
ma(5/4))

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GIAC/XCAS [A]  time = 0.233975, size = 278, normalized size = 4.88 \[ \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{4 \, b} + \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{4 \, b} - \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{8 \, b} + \frac{\sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{8 \, b} \]

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)*(-b)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1
/4)/x)/(-b)^(1/4))/b + 1/4*sqrt(2)*(-b)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^
(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b - 1/8*sqrt(2)*(-b)^(3/4)*ln(sqrt(-b
) + sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b + 1/8*sqrt(2
)*(-b)^(3/4)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 +
a)/x^2)/b