Optimal. Leaf size=275 \[ \frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{1}{a x} \]
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Rubi [A] time = 0.228284, antiderivative size = 275, 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, Rules used = {325, 301, 297, 1162, 617, 204, 1165, 628, 298, 205, 208} \[ \frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 301
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^8\right )} \, dx &=-\frac{1}{a x}-\frac{b \int \frac{x^6}{a+b x^8} \, dx}{a}\\ &=-\frac{1}{a x}+\frac{\sqrt{b} \int \frac{x^2}{\sqrt{-a}-\sqrt{b} x^4} \, dx}{2 a}-\frac{\sqrt{b} \int \frac{x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{2 a}\\ &=-\frac{1}{a x}+\frac{\sqrt [4]{b} \int \frac{1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 a}-\frac{\sqrt [4]{b} \int \frac{1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 a}+\frac{\sqrt [4]{b} \int \frac{\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 a}-\frac{\sqrt [4]{b} \int \frac{\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt{-a}+\sqrt{b} x^4} \, dx}{4 a}\\ &=-\frac{1}{a x}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 a}-\frac{\int \frac{1}{\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 a}+\frac{\sqrt [8]{b} \int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{b}}+2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \int \frac{\frac{\sqrt{2} \sqrt [8]{-a}}{\sqrt [8]{b}}-2 x}{-\frac{\sqrt [4]{-a}}{\sqrt [4]{b}}+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt{2} (-a)^{9/8}}\\ &=-\frac{1}{a x}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}\\ &=-\frac{1}{a x}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt [4]{-a}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt [4]{-a}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}\\ \end{align*}
Mathematica [A] time = 0.139074, size = 377, normalized size = 1.37 \[ -\frac{\sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+2 \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{a}}{8 a^{9/8} x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.004, size = 36, normalized size = 0.1 \begin{align*} -{\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}}}}-{\frac{1}{ax}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{1}{16} \,{\left (\frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{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 )}{a} + \frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{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 )}{a} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{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 )}{a} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{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 )}{a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \log \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 )}{a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \log \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 )}{a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \log \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 )}{a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{8}} \log \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 )}{a}\right )} b}{a} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.41967, size = 1013, normalized size = 3.68 \begin{align*} \frac{4 \, \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + \sqrt{2} a \sqrt{-\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} - b x^{2}}{b}} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} - 1\right ) + 4 \, \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + \sqrt{2} a \sqrt{\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}}{b}} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + 1\right ) - \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}\right ) + \sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} + b x^{2}\right ) + 8 \, a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a \sqrt{-\frac{a^{7} \left (-\frac{b}{a^{9}}\right )^{\frac{3}{4}} - b x^{2}}{b}} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}}\right ) - 2 \, a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + b x\right ) + 2 \, a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + b x\right ) - 16}{16 \, a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.402567, size = 29, normalized size = 0.11 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log{\left (- \frac{2097152 t^{7} a^{8}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27448, size = 601, normalized size = 2.19 \begin{align*} -\frac{b \sqrt{\sqrt{2} + 2} \left (\frac{a}{b}\right )^{\frac{7}{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{7}{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{7}{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{7}{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{7}{8}} \log \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{7}{8}} \log \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{7}{8}} \log \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{7}{8}} \log \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}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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