Integrand size = 23, antiderivative size = 61 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=-\frac {2 b}{7 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {8 b}{21 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \]
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Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2664, 2658} \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=-\frac {8 b}{21 f \sin ^{\frac {3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {2 b}{7 f \sin ^{\frac {7}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]
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Rule 2658
Rule 2664
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b}{7 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}+\frac {4}{7} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {5}{2}}(e+f x)} \, dx \\ & = -\frac {2 b}{7 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {8 b}{21 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\frac {2 b (-5+2 \cos (2 (e+f x)))}{21 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)} \]
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Time = 0.63 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {\frac {8 \left (\cos ^{3}\left (f x +e \right )\right )}{21}-\frac {2 \cos \left (f x +e \right )}{3}}{f \sin \left (f x +e \right )^{\frac {7}{2}} \sqrt {b \sec \left (f x +e \right )}}\) | \(43\) |
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none
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\frac {2 \, {\left (4 \, \cos \left (f x + e\right )^{4} - 7 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{21 \, {\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\text {Timed out} \]
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Time = 2.41 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx=\frac {4\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}\,\left (11\,\sin \left (e+f\,x\right )+4\,\sin \left (3\,e+3\,f\,x\right )-6\,\sin \left (5\,e+5\,f\,x\right )+\sin \left (7\,e+7\,f\,x\right )\right )}{21\,b\,f\,\sqrt {\sin \left (e+f\,x\right )}\,\left (15\,\cos \left (2\,e+2\,f\,x\right )-6\,\cos \left (4\,e+4\,f\,x\right )+\cos \left (6\,e+6\,f\,x\right )-10\right )} \]
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