Optimal. Leaf size=272 \[ -\frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{x}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.270615, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.333, Rules used = {193, 321, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [8]{b} \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{x}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 193
Rule 321
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+\frac{b}{x^8}} \, dx &=\int \frac{x^8}{b+a x^8} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^8} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt{b} \int \frac{1}{\sqrt{b}-\sqrt{-a} x^4} \, dx}{2 a}-\frac{\sqrt{b} \int \frac{1}{\sqrt{b}+\sqrt{-a} x^4} \, dx}{2 a}\\ &=\frac{x}{a}-\frac{\sqrt [4]{b} \int \frac{1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2} \, dx}{4 a}-\frac{\sqrt [4]{b} \int \frac{1}{\sqrt [4]{b}+\sqrt [4]{-a} x^2} \, dx}{4 a}-\frac{\sqrt [4]{b} \int \frac{\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt{b}+\sqrt{-a} x^4} \, dx}{4 a}-\frac{\sqrt [4]{b} \int \frac{\sqrt [4]{b}+\sqrt [4]{-a} x^2}{\sqrt{b}+\sqrt{-a} x^4} \, dx}{4 a}\\ &=\frac{x}{a}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \int \frac{\frac{\sqrt{2} \sqrt [8]{b}}{\sqrt [8]{-a}}+2 x}{-\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}-x^2} \, dx}{8 \sqrt{2} (-a)^{9/8}}-\frac{\sqrt [8]{b} \int \frac{\frac{\sqrt{2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{-\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}-x^2} \, dx}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [4]{b} \int \frac{1}{\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}-\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+x^2} \, dx}{8 (-a)^{5/4}}+\frac{\sqrt [4]{b} \int \frac{1}{\frac{\sqrt [4]{b}}{\sqrt [4]{-a}}+\frac{\sqrt{2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+x^2} \, dx}{8 (-a)^{5/4}}\\ &=\frac{x}{a}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt [4]{b}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} 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]{-a} x}{\sqrt [8]{b}}\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]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}\\ &=\frac{x}{a}+\frac{\sqrt [8]{b} \tan ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \tanh ^{-1}\left (\frac{\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac{\sqrt [8]{b} \log \left (\sqrt [4]{b}-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}+\frac{\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt{2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt{2} (-a)^{9/8}}\\ \end{align*}
Mathematica [A] time = 0.231781, size = 367, normalized size = 1.35 \[ \frac{\sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )+\sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-2 \sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac{\pi }{8}\right )\right )-2 \sqrt [8]{b} \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{a} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}\right )-2 \sqrt [8]{b} \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{a} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{b}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [8]{a} x}{8 a^{9/8}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.015, size = 34, normalized size = 0.1 \begin{align*}{\frac{x}{a}}-{\frac{b}{8\,{a}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+b \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{1}{16} \, b{\left (\frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{b} + \frac{2 \, \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{b} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{b} + \frac{2 \, \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{b} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{b} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{b} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{b} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{b}\right )}}{a} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.66395, size = 972, normalized size = 3.57 \begin{align*} -\frac{4 \, \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - \sqrt{2} \sqrt{\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}} a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - b}{b}\right ) + 4 \, \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{\sqrt{2} a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - \sqrt{2} \sqrt{-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}} a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} + b}{b}\right ) + \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) - \sqrt{2} a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-\sqrt{2} a x \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}\right ) + 8 \, a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{8} x \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}} - \sqrt{a^{2} \left (-\frac{b}{a^{9}}\right )^{\frac{1}{4}} + x^{2}} a^{8} \left (-\frac{b}{a^{9}}\right )^{\frac{7}{8}}}{b}\right ) + 2 \, a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + x\right ) - 2 \, a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} \log \left (-a \left (-\frac{b}{a^{9}}\right )^{\frac{1}{8}} + x\right ) - 16 \, x}{16 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.348609, size = 22, normalized size = 0.08 \begin{align*} \operatorname{RootSum}{\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log{\left (- 8 t a + x \right )} \right )\right )} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.30222, size = 586, normalized size = 2.15 \begin{align*} -\frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} \left (\frac{b}{a}\right )^{\frac{1}{8}} + \left (\frac{b}{a}\right )^{\frac{1}{4}}\right )}{16 \, a} + \frac{x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]