3.1083 \(\int \frac{1}{x \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=55 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

[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 = 0.0857372, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}} \]

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 = 9.37665, size = 46, normalized size = 0.84 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a}} \]

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 = 0.0333406, size = 46, normalized size = 0.84 \[ -\frac{\sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )}{\sqrt [4]{a+b x^4}} \]

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 = 0.028, 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 = 0.281615, size = 105, normalized size = 1.91 \[ -\frac{\arctan \left (\frac{a^{\frac{1}{4}}}{\sqrt{\sqrt{b x^{4} + a} + \sqrt{a}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right )}{a^{\frac{1}{4}}} - \frac{\log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} + \frac{\log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right )}{4 \, a^{\frac{1}{4}}} \]

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 = 3.62277, size = 37, normalized size = 0.67 \[ - \frac{\Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231199, size = 251, normalized size = 4.56 \[ -\frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \]

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