Integrand size = 20, antiderivative size = 47 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]
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Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]
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Rule 3396
Rubi steps \begin{align*} \text {integral}& = \frac {3}{5} \int x \sqrt {\cos (x)} \, dx+\int \frac {x}{\cos ^{\frac {7}{2}}(x)} \, dx \\ & = -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {3}{5} \int \frac {x}{\cos ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\cos (x)} \, dx \\ & = -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\frac {46 \cos (x)+18 \cos (3 x)+21 x \sin (x)+9 x \sin (3 x)}{30 \cos ^{\frac {5}{2}}(x)} \]
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\[\int \left (\frac {x}{\cos \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cos }\left (x \right )\right )}{5}\right )d x\]
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Exception generated. \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\text {Timed out} \]
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\[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cos \left (x\right )} + \frac {x}{\cos \left (x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cos \left (x\right )} + \frac {x}{\cos \left (x\right )^{\frac {7}{2}}} \,d x } \]
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Time = 13.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\frac {36\,{\cos \left (x\right )}^3+18\,x\,\sin \left (x\right )\,{\cos \left (x\right )}^2-4\,\cos \left (x\right )+6\,x\,\sin \left (x\right )}{15\,{\cos \left (x\right )}^{5/2}} \]
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