Integrand size = 7, antiderivative size = 85 \[ \int \cos (x) \sec (6 x) \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]
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Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4441, 2082, 213, 1180} \[ \int \cos (x) \sec (6 x) \, dx=-\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \]
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Rule 213
Rule 1180
Rule 2082
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-18 x^2+48 x^4-32 x^6} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{3 \left (-1+2 x^2\right )}-\frac {4 \left (-1+2 x^2\right )}{3 \left (1-16 x^2+16 x^4\right )}\right ) \, dx,x,\sin (x)\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\sin (x)\right )-\frac {4}{3} \text {Subst}\left (\int \frac {-1+2 x^2}{1-16 x^2+16 x^4} \, dx,x,\sin (x)\right ) \\ & = -\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-8-4 \sqrt {3}+16 x^2} \, dx,x,\sin (x)\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1}{-8+4 \sqrt {3}+16 x^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {\text {arctanh}\left (\sqrt {2} \sin (x)\right )}{3 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )}{6 \sqrt {2-\sqrt {3}}}+\frac {\text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )}{6 \sqrt {2+\sqrt {3}}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \cos (x) \sec (6 x) \, dx=\frac {1}{6} \left (-\sqrt {2} \text {arctanh}\left (\sqrt {2} \sin (x)\right )+\sqrt {2+\sqrt {3}} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2-\sqrt {3}}}\right )+\sqrt {2-\sqrt {3}} \text {arctanh}\left (\frac {2 \sin (x)}{\sqrt {2+\sqrt {3}}}\right )\right ) \]
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Time = 1.81 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{6}+\frac {2 \,\operatorname {arctanh}\left (\frac {8 \sin \left (x \right )}{2 \sqrt {6}+2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}+2 \sqrt {2}\right )}+\frac {2 \,\operatorname {arctanh}\left (\frac {8 \sin \left (x \right )}{2 \sqrt {6}-2 \sqrt {2}}\right )}{3 \left (2 \sqrt {6}-2 \sqrt {2}\right )}\) | \(80\) |
risch | \(\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{12}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{12}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (331776 \textit {\_Z}^{4}-2304 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-13824 i \textit {\_R}^{3}+96 i \textit {\_R} \right ) {\mathrm e}^{i x}-1\right )\right )\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (67) = 134\).
Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.81 \[ \int \cos (x) \sec (6 x) \, dx=-\frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} + 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {\sqrt {3} + 2} \log \left (\sqrt {\sqrt {3} + 2} {\left (\sqrt {3} - 2\right )} - 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} + 2 \, \sin \left (x\right )\right ) - \frac {1}{12} \, \sqrt {-\sqrt {3} + 2} \log \left ({\left (\sqrt {3} + 2\right )} \sqrt {-\sqrt {3} + 2} - 2 \, \sin \left (x\right )\right ) + \frac {1}{12} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) \]
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\[ \int \cos (x) \sec (6 x) \, dx=\int \cos {\left (x \right )} \sec {\left (6 x \right )}\, dx \]
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\[ \int \cos (x) \sec (6 x) \, dx=\int { \cos \left (x\right ) \sec \left (6 \, x\right ) \,d x } \]
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none
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.55 \[ \int \cos (x) \sec (6 x) \, dx=\frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left ({\left | \frac {1}{4} \, \sqrt {6} + \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left ({\left | \frac {1}{4} \, \sqrt {6} - \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left ({\left | -\frac {1}{4} \, \sqrt {6} + \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left ({\left | -\frac {1}{4} \, \sqrt {6} - \frac {1}{4} \, \sqrt {2} + \sin \left (x\right ) \right |}\right ) + \frac {1}{12} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) \]
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Time = 26.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \cos (x) \sec (6 x) \, dx=\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}+\frac {3\,\sqrt {6}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}+\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}+\frac {\sqrt {6}}{12}\right )-\mathrm {atanh}\left (\frac {5\,\sqrt {2}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}-\frac {3\,\sqrt {6}\,\sin \left (x\right )}{2097152\,\left (\frac {\sqrt {2}\,\sqrt {6}}{4194304}-\frac {1}{1048576}\right )}\right )\,\left (\frac {\sqrt {2}}{12}-\frac {\sqrt {6}}{12}\right )-\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{6} \]
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