Integrand size = 9, antiderivative size = 335 \[ \int \frac {1}{a+b x^7} \, dx=\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {2 \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
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Time = 0.49 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {207, 648, 632, 210, 642, 31} \[ \int \frac {1}{a+b x^7} \, dx=\frac {2 \cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \cos \left (\frac {\pi }{14}\right ) \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {2 \sin \left (\frac {\pi }{7}\right ) \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\sin \left (\frac {3 \pi }{14}\right ) \log \left (a^{2/7}+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\sin \left (\frac {\pi }{14}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )+b^{2/7} x^2\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}} \]
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Rule 31
Rule 207
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )} \, dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{a}-\sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )} \, dx}{7 a^{6/7}}+\frac {2 \int \frac {\sqrt [7]{a}+\sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )} \, dx}{7 a^{6/7}}+\frac {\int \frac {1}{\sqrt [7]{a}+\sqrt [7]{b} x} \, dx}{7 a^{6/7}} \\ & = \frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\left (2 \cos ^2\left (\frac {\pi }{14}\right )\right ) \int \frac {1}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )} \, dx}{7 a^{5/7}}-\frac {\cos \left (\frac {\pi }{7}\right ) \int \frac {2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )} \, dx}{7 a^{6/7} \sqrt [7]{b}}+\frac {\left (2 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \int \frac {1}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )} \, dx}{7 a^{5/7}}-\frac {\sin \left (\frac {\pi }{14}\right ) \int \frac {2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )} \, dx}{7 a^{6/7} \sqrt [7]{b}}+\frac {\left (2 \sin ^2\left (\frac {\pi }{7}\right )\right ) \int \frac {1}{a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )} \, dx}{7 a^{5/7}}+\frac {\sin \left (\frac {3 \pi }{14}\right ) \int \frac {2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )}{a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )} \, dx}{7 a^{6/7} \sqrt [7]{b}} \\ & = \frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\left (4 \cos ^2\left (\frac {\pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac {\pi }{14}\right )} \, dx,x,2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {\pi }{14}\right )\right )}{7 a^{5/7}}-\frac {\left (4 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 a^{2/7} b^{2/7} \cos ^2\left (\frac {3 \pi }{14}\right )} \, dx,x,2 b^{2/7} x+2 \sqrt [7]{a} \sqrt [7]{b} \sin \left (\frac {3 \pi }{14}\right )\right )}{7 a^{5/7}}-\frac {\left (4 \sin ^2\left (\frac {\pi }{7}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 a^{2/7} b^{2/7} \sin ^2\left (\frac {\pi }{7}\right )} \, dx,x,2 b^{2/7} x-2 \sqrt [7]{a} \sqrt [7]{b} \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{5/7}} \\ & = \frac {2 \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}}-\frac {2 \tan ^{-1}\left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )}{7 a^{6/7} \sqrt [7]{b}}+\frac {\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.78 \[ \int \frac {1}{a+b x^7} \, dx=\frac {2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {\pi }{14}\right )}{\sqrt [7]{a}}-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )+2 \arctan \left (\frac {\sqrt [7]{b} x \sec \left (\frac {3 \pi }{14}\right )}{\sqrt [7]{a}}+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\log \left (\sqrt [7]{a}+\sqrt [7]{b} x\right )-\cos \left (\frac {\pi }{7}\right ) \log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \cos \left (\frac {\pi }{7}\right )\right )-\log \left (a^{2/7}+b^{2/7} x^2-2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-2 \arctan \left (\cot \left (\frac {\pi }{7}\right )-\frac {\sqrt [7]{b} x \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{a}}\right ) \sin \left (\frac {\pi }{7}\right )+\log \left (a^{2/7}+b^{2/7} x^2+2 \sqrt [7]{a} \sqrt [7]{b} x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )}{7 a^{6/7} \sqrt [7]{b}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 8.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.08
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{7}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\) | \(27\) |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{7}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{6}}}{7 b}\) | \(27\) |
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Exception generated. \[ \int \frac {1}{a+b x^7} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.06 \[ \int \frac {1}{a+b x^7} \, dx=\operatorname {RootSum} {\left (823543 t^{7} a^{6} b - 1, \left ( t \mapsto t \log {\left (7 t a + x \right )} \right )\right )} \]
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\[ \int \frac {1}{a+b x^7} \, dx=\int { \frac {1}{b x^{7} + a} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+b x^7} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) \log \left (-2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) \log \left (2 \, x \left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x^{2} + \left (-\frac {a}{b}\right )^{\frac {2}{7}}\right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {3}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {2}{7} \, \pi \right ) - x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right )}{7 \, a} + \frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{7}} \arctan \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \cos \left (\frac {1}{7} \, \pi \right ) + x}{\left (-\frac {a}{b}\right )^{\frac {1}{7}} \sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right )}{7 \, a} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{7}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{7}} \right |}\right )}{7 \, a} \]
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Time = 6.70 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.59 \[ \int \frac {1}{a+b x^7} \, dx=\frac {\ln \left (b^{1/7}\,x+a^{1/7}\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (b^{1/7}\,x-a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7\,a^{6/7}\,b^{1/7}}+\frac {{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\,\ln \left (a^{1/7}\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}+b^{1/7}\,x\right )}{7\,a^{6/7}\,b^{1/7}} \]
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