3.1113 \(\int \frac{1}{x \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=55 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}} \]

[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 = 0.0781424, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}} \]

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 = 8.2933, size = 48, normalized size = 0.87 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}}} \]

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 = 0.0351953, size = 48, normalized size = 0.87 \[ -\frac{\left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )}{3 \left (a+b x^4\right )^{3/4}} \]

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 = 0.031, 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 = 0.25846, size = 134, 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 = 3.80776, size = 39, normalized size = 0.71 \[ - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220474, size = 251, normalized size = 4.56 \[ -\frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a} \]

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