3.1207 \(\int \frac{1}{x \sqrt [4]{a-b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0899578, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^4)^(1/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.918, size = 46, normalized size = 0.81 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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**(1/4))

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Mathematica [C]  time = 0.037766, size = 47, normalized size = 0.82 \[ -\frac{\sqrt [4]{1-\frac{a}{b x^4}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{a}{b x^4}\right )}{\sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-(((1 - a/(b*x^4))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, a/(b*x^4)])/(a - b*x^4
)^(1/4))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^4+a)^(1/4),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249457, size = 111, normalized size = 1.95 \[ -\frac{\arctan \left (\frac{a^{\frac{1}{4}}}{\sqrt{\sqrt{-b x^{4} + a} + \sqrt{a}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\right )}{a^{\frac{1}{4}}} - \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} + \frac{\log \left ({\left (-b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.8113, size = 39, normalized size = 0.68 \[ - \frac{e^{- \frac{i \pi }{4}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-exp(-I*pi/4)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), a/(b*x**4))/(4*b**(1/4)*x*gam
ma(5/4))

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GIAC/XCAS [A]  time = 0.226822, size = 259, normalized size = 4.54 \[ -\frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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