Integrand size = 9, antiderivative size = 267 \[ \int \frac {1}{a+b x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}} \]
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Time = 0.13 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{a+b x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{7/8} \sqrt [8]{b}}+\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{7/8} \sqrt [8]{b}} \]
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Rule 210
Rule 211
Rule 214
Rule 217
Rule 218
Rule 220
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = -\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 {\text {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 {\text {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*}
Time = 0.09 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.21 \[ \int \frac {1}{a+b x^8} \, dx=\frac {2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-2 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+2 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 a^{7/8} \sqrt [8]{b}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 11.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.10
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(27\) |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\) | \(27\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+b x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} i \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (i \, a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} i \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-i \, a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} \, \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.07 \[ \int \frac {1}{a+b x^8} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{7} b + 1, \left ( t \mapsto t \log {\left (8 t a + x \right )} \right )\right )} \]
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\[ \int \frac {1}{a+b x^8} \, dx=\int { \frac {1}{b x^{8} + a} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (182) = 364\).
Time = 0.29 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.64 \[ \int \frac {1}{a+b x^8} \, dx=\frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{b}\right )^{\frac {1}{8}} + \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} \]
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Time = 6.33 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.41 \[ \int \frac {1}{a+b x^8} \, dx=-\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{{\left (-a\right )}^{1/8}}\right )}{4\,{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{{\left (-a\right )}^{7/8}\,b^{1/8}} \]
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