Optimal. Leaf size=277 \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.510975, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.846 \[ -\frac{b^{3/8} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{b^{3/8} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{11/8}}+\frac{b^{3/8} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{11/8}}-\frac{b^{3/8} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x^8)),x]
[Out]
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Rubi in Sympy [A] time = 109.79, size = 255, normalized size = 0.92 \[ - \frac{\sqrt{2} b^{\frac{3}{8}} \log{\left (- \sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \log{\left (\sqrt{2} \sqrt [8]{b} x \sqrt [8]{- a} + \sqrt [4]{b} x^{2} + \sqrt [4]{- a} \right )}}{16 \left (- a\right )^{\frac{11}{8}}} - \frac{b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} - 1 \right )}}{8 \left (- a\right )^{\frac{11}{8}}} + \frac{\sqrt{2} b^{\frac{3}{8}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{- a}} + 1 \right )}}{8 \left (- a\right )^{\frac{11}{8}}} - \frac{b^{\frac{3}{8}} \operatorname{atanh}{\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{- a}} \right )}}{4 \left (- a\right )^{\frac{11}{8}}} - \frac{1}{3 a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x**8+a),x)
[Out]
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Mathematica [A] time = 0.365373, size = 395, normalized size = 1.43 \[ \frac{-8 a^{3/8}+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-6 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+3 b^{3/8} x^3 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{24 a^{11/8} x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x^8)),x]
[Out]
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Maple [C] time = 0.007, size = 36, normalized size = 0.1 \[ -{\frac{1}{8\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}}-{\frac{1}{3\,a{x}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^8+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{x^{4}}{b x^{8} + a}\,{d x}}{a} - \frac{1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241778, size = 698, normalized size = 2.52 \[ -\frac{\sqrt{2}{\left (12 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{b^{2} x + b^{2} \sqrt{-\frac{a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b x^{2}}{b}}}\right ) - 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) + 3 \, \sqrt{2} a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + b^{2} x\right ) - 12 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + \sqrt{2} b^{2} x + \sqrt{2} b^{2} \sqrt{\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}}{b^{2}}}}\right ) - 12 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}}}{a^{7} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - \sqrt{2} b^{2} x - \sqrt{2} b^{2} \sqrt{-\frac{\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} + a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} - b^{2} x^{2}}{b^{2}}}}\right ) + 3 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) - 3 \, a x^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{7} x \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{5}{8}} - a^{3} b \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + b^{2} x^{2}\right ) + 8 \, \sqrt{2}\right )}}{48 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.50367, size = 36, normalized size = 0.13 \[ \operatorname{RootSum}{\left (16777216 t^{8} a^{11} + b^{3}, \left ( t \mapsto t \log{\left (\frac{32768 t^{5} a^{7}}{b^{2}} + x \right )} \right )\right )} - \frac{1}{3 a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x**8+a),x)
[Out]
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GIAC/XCAS [A] time = 0.257668, size = 601, normalized size = 2.17 \[ \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}}}\right )}{8 \, a^{2}} + \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} + \frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{5}{8}}{\rm ln}\left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{1}{8}} + \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}{16 \, a^{2}} - \frac{1}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^8 + a)*x^4),x, algorithm="giac")
[Out]