Optimal. Leaf size=267 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{3/8} b^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{3/8} b^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{3/8} b^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{3/8} b^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}} \]
[Out]
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Rubi [A] time = 0.41934, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.769 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{3/8} b^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{3/8} b^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{3/8} b^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{3/8} b^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{3/8} b^{5/8}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^8),x]
[Out]
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Rubi in Sympy [A] time = 93.9288, size = 246, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 b^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**8+a),x)
[Out]
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Mathematica [A] time = 0.446088, size = 324, normalized size = 1.21 \[ -\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 a^{3/8} b^{5/8}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^8),x]
[Out]
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Maple [C] time = 0.018, size = 27, normalized size = 0.1 \[{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^8+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{b x^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^8 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240745, size = 595, normalized size = 2.23 \[ \frac{1}{16} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}}}{x + \sqrt{-a b \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - \sqrt{2} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \log \left (a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} + x\right ) + \sqrt{2} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \log \left (-a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} + x\right ) - 4 \, \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}}}{a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} + \sqrt{2} x + \sqrt{2} \sqrt{\sqrt{2} a^{2} b^{3} x \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} - a b \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{4}} + x^{2}}}\right ) - 4 \, \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}}}{a^{2} b^{3} \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} - \sqrt{2} x - \sqrt{2} \sqrt{-\sqrt{2} a^{2} b^{3} x \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} - a b \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{4}} + x^{2}}}\right ) + \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{2} b^{3} x \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} - a b \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{4}} + x^{2}\right ) - \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{2} b^{3} x \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{5}{8}} - a b \left (-\frac{1}{a^{3} b^{5}}\right )^{\frac{1}{4}} + x^{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^8 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.499208, size = 29, normalized size = 0.11 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{3} b^{5} + 1, \left ( t \mapsto t \log{\left (- 32768 t^{5} a^{2} b^{3} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**8+a),x)
[Out]
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GIAC/XCAS [A] time = 0.256302, size = 579, normalized size = 2.17 \[ -\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^8 + a),x, algorithm="giac")
[Out]