Integrand size = 66, antiderivative size = 114 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {\pi -2 k \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )+2 \arctan \left (\left (\frac {a}{b}\right )^{-1/n} \left (x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 k \pi }{n}\right )\right ) \csc \left (\frac {\pi -2 k \pi }{n}\right )\right ) \sin \left (\frac {\pi -2 k \pi }{n}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {12, 648, 632, 210, 642} \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=2 \sin \left (\frac {\pi -2 \pi k}{n}\right ) \arctan \left (\left (\frac {a}{b}\right )^{-1/n} \csc \left (\frac {\pi -2 \pi k}{n}\right ) \left (x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 \pi k}{n}\right )\right )\right )-\cos \left (\frac {\pi -2 \pi k}{n}\right ) \log \left (-2 x \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi -2 \pi k}{n}\right )+\left (\frac {a}{b}\right )^{2/n}+x^2\right ) \]
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Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx \\ & = -\left (\cos \left (\frac {(-1+2 k) \pi }{n}\right ) \int \frac {2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {(-1+2 k) \pi }{n}\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx\right )+\left (2 \left (\frac {a}{b}\right )^{\frac {1}{n}}-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos ^2\left (\frac {(-1+2 k) \pi }{n}\right )\right ) \int \frac {1}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx \\ & = -\cos \left (\frac {(1-2 k) \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )+\left (2 \left (-2 \left (\frac {a}{b}\right )^{\frac {1}{n}}+2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos ^2\left (\frac {(-1+2 k) \pi }{n}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \left (\frac {a}{b}\right )^{2/n} \sin ^2\left (\frac {(1-2 k) \pi }{n}\right )} \, dx,x,2 x-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right ) \\ & = -\cos \left (\frac {(1-2 k) \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi -2 k \pi }{n}\right )\right )+2 \tan ^{-1}\left (\left (\frac {a}{b}\right )^{-1/n} \left (x-\left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {(1-2 k) \pi }{n}\right )\right ) \csc \left (\frac {\pi -2 k \pi }{n}\right )\right ) \csc \left (\frac {\pi -2 k \pi }{n}\right ) \sin ^2\left (\frac {(1-2 k) \pi }{n}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=2 \left (-\frac {1}{2} \cos \left (\frac {(-1+2 k) \pi }{n}\right ) \log \left (\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )-\arctan \left (\cot \left (\frac {(-1+2 k) \pi }{n}\right )-\left (\frac {a}{b}\right )^{-1/n} x \csc \left (\frac {(-1+2 k) \pi }{n}\right )\right ) \sin \left (\frac {(-1+2 k) \pi }{n}\right )\right ) \]
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Time = 22.88 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.79
method | result | size |
default | \(-\cos \left (\frac {\pi \left (2 k -1\right )}{n}\right ) \ln \left (2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )-x^{2}-\left (\frac {a}{b}\right )^{\frac {2}{n}}\right )+\frac {2 \left (\left (\cos ^{2}\left (\frac {\pi \left (2 k -1\right )}{n}\right )\right ) \left (\frac {a}{b}\right )^{\frac {1}{n}}-\left (\frac {a}{b}\right )^{\frac {1}{n}}\right ) \arctan \left (\frac {2 \left (\frac {a}{b}\right )^{\frac {1}{n}} \cos \left (\frac {\pi \left (2 k -1\right )}{n}\right )-2 x}{2 \sqrt {-\left (\frac {a}{b}\right )^{\frac {2}{n}} \left (\cos ^{2}\left (\frac {\pi \left (2 k -1\right )}{n}\right )\right )+\left (\frac {a}{b}\right )^{\frac {2}{n}}}}\right )}{\sqrt {-\left (\frac {a}{b}\right )^{\frac {2}{n}} \left (\cos ^{2}\left (\frac {\pi \left (2 k -1\right )}{n}\right )\right )+\left (\frac {a}{b}\right )^{\frac {2}{n}}}}\) | \(204\) |
risch | \(\text {Expression too large to display}\) | \(860\) |
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Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \log \left (-\frac {2 \, {\left (2 \, x \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x^{2} - \left (\frac {a}{b}\right )^{\frac {2}{n}}\right )}}{\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) + 1}\right ) - 2 \, \arctan \left (\frac {\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x}{\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \sin \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )}\right ) \sin \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \]
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Time = 0.46 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=- \left (- \sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \log {\left (x - \left (\frac {a}{b}\right )^{\frac {1}{n}} \left (- \sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \right )} - \left (\sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \log {\left (x - \left (\frac {a}{b}\right )^{\frac {1}{n}} \left (\sqrt {\left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} - 1\right ) \left (\cos {\left (\frac {\pi \left (2 k - 1\right )}{n} \right )} + 1\right )} + \cos {\left (\frac {2 \pi k}{n} - \frac {\pi }{n} \right )}\right ) \right )} \]
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Exception generated. \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.77 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) \log \left (-2 \, x \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) + x^{2} + \left (\frac {a}{b}\right )^{\frac {2}{n}}\right ) - \frac {2 \, {\left (\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} - \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}\right )} \arctan \left (-\frac {\left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )} \cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right ) - x}{\sqrt {-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} + 1} \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}}\right )}{\sqrt {-\cos \left (\frac {2 \, \pi k}{n} - \frac {\pi }{n}\right )^{2} + 1} \left (\frac {a}{b}\right )^{\left (\frac {1}{n}\right )}} \]
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Time = 10.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.56 \[ \int \frac {2 \left (\left (\frac {a}{b}\right )^{\frac {1}{n}}-x \cos \left (\frac {(-1+2 k) \pi }{n}\right )\right )}{\left (\frac {a}{b}\right )^{2/n}+x^2-2 \left (\frac {a}{b}\right )^{\frac {1}{n}} x \cos \left (\frac {(-1+2 k) \pi }{n}\right )} \, dx=-2\,\mathrm {atan}\left (\frac {2\,x\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}-2\,\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,{\left (\frac {a}{b}\right )}^{1/n}\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}}{2\,{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2\,{\left (\frac {a}{b}\right )}^{1/n}-2\,{\left (\frac {a}{b}\right )}^{1/n}}\right )\,\sqrt {1-{\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )}^2}-\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,\ln \left ({\left (\frac {a}{b}\right )}^{2/n}+x^2-2\,x\,\cos \left (\frac {\Pi \,\left (2\,k-1\right )}{n}\right )\,{\left (\frac {a}{b}\right )}^{1/n}\right ) \]
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