Optimal. Leaf size=239 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}} \]
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Rubi [A] time = 0.168881, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}} \]
Antiderivative was successfully verified.
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Rule 214
Rule 212
Rule 208
Rule 205
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a-b x^8} \, dx &=\frac{\int \frac{1}{\sqrt{a}-\sqrt{b} x^4} \, dx}{2 \sqrt{a}}+\frac{\int \frac{1}{\sqrt{a}+\sqrt{b} x^4} \, dx}{2 \sqrt{a}}\\ &=\frac{\int \frac{1}{\sqrt [4]{a}-\sqrt [4]{b} x^2} \, dx}{4 a^{3/4}}+\frac{\int \frac{1}{\sqrt [4]{a}+\sqrt [4]{b} x^2} \, dx}{4 a^{3/4}}+\frac{\int \frac{\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt{a}+\sqrt{b} x^4} \, dx}{4 a^{3/4}}+\frac{\int \frac{\sqrt [4]{a}+\sqrt [4]{b} x^2}{\sqrt{a}+\sqrt{b} x^4} \, dx}{4 a^{3/4}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\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^{3/4} \sqrt [4]{b}}+\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^{3/4} \sqrt [4]{b}}-\frac{\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^{7/8} \sqrt [8]{b}}-\frac{\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^{7/8} \sqrt [8]{b}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\log \left (\sqrt [4]{a}-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt [4]{a}+\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\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^{7/8} \sqrt [8]{b}}-\frac{\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^{7/8} \sqrt [8]{b}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac{\log \left (\sqrt [4]{a}-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}+\frac{\log \left (\sqrt [4]{a}+\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt{2} a^{7/8} \sqrt [8]{b}}\\ \end{align*}
Mathematica [A] time = 0.0563742, size = 198, normalized size = 0.83 \[ \frac{-\sqrt{2} \log \left (-\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \log \left (\sqrt [8]{a}-\sqrt [8]{b} x\right )+2 \log \left (\sqrt [8]{a}+\sqrt [8]{b} x\right )+4 \tan ^{-1}\left (\frac{\sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{16 a^{7/8} \sqrt [8]{b}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 29, normalized size = 0.1 \begin{align*} -{\frac{1}{8\,b}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{8}-a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \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*} -\int \frac{1}{b x^{8} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39707, size = 1056, normalized size = 4.42 \begin{align*} -\frac{1}{4} \, \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (-\sqrt{2} a^{6} b x \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}} + \sqrt{2} \sqrt{\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}} a^{6} b \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}} - 1\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (-\sqrt{2} a^{6} b x \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}} + \sqrt{2} \sqrt{-\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}} a^{6} b \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}} + 1\right ) + \frac{1}{16} \, \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right ) - \frac{1}{16} \, \sqrt{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a x \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}\right ) - \frac{1}{2} \, \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \arctan \left (-a^{6} b x \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}} + \sqrt{a^{2} \left (\frac{1}{a^{7} b}\right )^{\frac{1}{4}} + x^{2}} a^{6} b \left (\frac{1}{a^{7} b}\right )^{\frac{7}{8}}\right ) + \frac{1}{8} \, \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) - \frac{1}{8} \, \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} \log \left (-a \left (\frac{1}{a^{7} b}\right )^{\frac{1}{8}} + x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.175374, size = 22, normalized size = 0.09 \begin{align*} - \operatorname{RootSum}{\left (16777216 t^{8} a^{7} b - 1, \left ( t \mapsto t \log{\left (- 8 t a + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21413, size = 612, normalized size = 2.56 \begin{align*} \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{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{1}{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{1}{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{1}{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{1}{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} - \frac{\sqrt{\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{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} + \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{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} - \frac{\sqrt{-\sqrt{2} + 2} \left (-\frac{a}{b}\right )^{\frac{1}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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