3.1237 \(\int \frac{1}{x \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}} \]

[Out]

-ArcTan[(a - b*x^4)^(1/4)/a^(1/4)]/(2*a^(3/4)) - ArcTanh[(a - b*x^4)^(1/4)/a^(1/
4)]/(2*a^(3/4))

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Rubi [A]  time = 0.0851296, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^4)^(3/4)),x]

[Out]

-ArcTan[(a - b*x^4)^(1/4)/a^(1/4)]/(2*a^(3/4)) - ArcTanh[(a - b*x^4)^(1/4)/a^(1/
4)]/(2*a^(3/4))

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Rubi in Sympy [A]  time = 9.30386, size = 48, normalized size = 0.84 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-b*x**4+a)**(3/4),x)

[Out]

-atan((a - b*x**4)**(1/4)/a**(1/4))/(2*a**(3/4)) - atanh((a - b*x**4)**(1/4)/a**
(1/4))/(2*a**(3/4))

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Mathematica [C]  time = 0.0366915, size = 49, normalized size = 0.86 \[ -\frac{\left (1-\frac{a}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{a}{b x^4}\right )}{3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a - b*x^4)^(3/4)),x]

[Out]

-((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 = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x/(-b*x^4+a)^(3/4),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)^(3/4)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241867, size = 139, normalized size = 2.44 \[ \frac{1}{a^{3}}^{\frac{1}{4}} \arctan \left (\frac{a \frac{1}{a^{3}}^{\frac{1}{4}}}{\sqrt{a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{-b x^{4} + a}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \frac{1}{a^{3}}^{\frac{1}{4}} \log \left (a \frac{1}{a^{3}}^{\frac{1}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right ) + \frac{1}{4} \, \frac{1}{a^{3}}^{\frac{1}{4}} \log \left (-a \frac{1}{a^{3}}^{\frac{1}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.10843, size = 42, normalized size = 0.74 \[ - \frac{e^{- \frac{3 i \pi }{4}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-b*x**4+a)**(3/4),x)

[Out]

-exp(-3*I*pi/4)*gamma(3/4)*hyper((3/4, 3/4), (7/4,), a/(b*x**4))/(4*b**(3/4)*x**
3*gamma(7/4))

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GIAC/XCAS [A]  time = 0.221382, size = 259, normalized size = 4.54 \[ -\frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*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))/a - 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))/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))/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))
/a