Integrand size = 20, antiderivative size = 24 \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \]
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Rule 4272
Rule 4274
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {3}{5} \int \frac {x}{\sqrt {\sec (x)}} \, dx\right )+\int \frac {x}{\sec ^{\frac {5}{2}}(x)} \, dx \\ & = \frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)}+\frac {3}{5} \int \frac {x}{\sqrt {\sec (x)}} \, dx-\frac {1}{5} \left (3 \sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int x \sqrt {\cos (x)} \, dx \\ & = \frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {2 (2+5 x \tan (x))}{25 \sec ^{\frac {5}{2}}(x)} \]
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\[\int \left (\frac {x}{\sec \left (x \right )^{\frac {5}{2}}}-\frac {3 x}{5 \sqrt {\sec \left (x \right )}}\right )d x\]
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Exception generated. \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=- \frac {\int \left (- \frac {5 x}{\sec ^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {3 x}{\sqrt {\sec {\left (x \right )}}}\, dx}{5} \]
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\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\sec \left (x\right )}} + \frac {x}{\sec \left (x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\sec \left (x\right )}} + \frac {x}{\sec \left (x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=-\int \frac {3\,x}{5\,\sqrt {\frac {1}{\cos \left (x\right )}}}-\frac {x}{{\left (\frac {1}{\cos \left (x\right )}\right )}^{5/2}} \,d x \]
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