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