3.1178 \(\int \frac{\sqrt [4]{a-b x^4}}{x} \, dx\)

Optimal. Leaf size=69 \[ \sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

(a - b*x^4)^(1/4) - (a^(1/4)*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/2 - (a^(1/4)*Arc
Tanh[(a - b*x^4)^(1/4)/a^(1/4)])/2

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Rubi [A]  time = 0.103215, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \sqrt [4]{a-b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^4)^(1/4)/x,x]

[Out]

(a - b*x^4)^(1/4) - (a^(1/4)*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/2 - (a^(1/4)*Arc
Tanh[(a - b*x^4)^(1/4)/a^(1/4)])/2

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Rubi in Sympy [A]  time = 11.24, size = 56, normalized size = 0.81 \[ - \frac{\sqrt [4]{a} \operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2} - \frac{\sqrt [4]{a} \operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2} + \sqrt [4]{a - b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0444338, size = 63, normalized size = 0.91 \[ \sqrt [4]{a-b x^4}-\frac{a \left (1-\frac{a}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{a}{b x^4}\right )}{3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^4)^(1/4)/x,x]

[Out]

(a - b*x^4)^(1/4) - (a*(1 - a/(b*x^4))^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, a/
(b*x^4)])/(3*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [4]{-b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^4+a)^(1/4)/x,x)

[Out]

int((-b*x^4+a)^(1/4)/x,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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269694, size = 123, normalized size = 1.78 \[ a^{\frac{1}{4}} \arctan \left (\frac{a^{\frac{1}{4}}}{\sqrt{\sqrt{-b x^{4} + a} + \sqrt{a}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right ) + \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right ) +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.02041, size = 44, normalized size = 0.64 \[ - \frac{\sqrt [4]{b} x e^{\frac{i \pi }{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(1/4)*x*exp(I*pi/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), a/(b*x**4))/(4*g
amma(3/4))

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GIAC/XCAS [A]  time = 0.22328, size = 257, normalized size = 3.72 \[ -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right ) +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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