Integrand size = 23, antiderivative size = 91 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=-\frac {2 b}{11 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {16 b}{77 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {64 b}{231 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \]
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Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2664, 2658} \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=-\frac {64 b}{231 f \sin ^{\frac {3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {16 b}{77 f \sin ^{\frac {7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {2 b}{11 f \sin ^{\frac {11}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \]
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Rule 2658
Rule 2664
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b}{11 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}+\frac {8}{11} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx \\ & = -\frac {2 b}{11 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {16 b}{77 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}+\frac {32}{77} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {5}{2}}(e+f x)} \, dx \\ & = -\frac {2 b}{11 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {16 b}{77 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {64 b}{231 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\frac {2 b (-45+28 \cos (2 (e+f x))-4 \cos (4 (e+f x)))}{231 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)} \]
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Time = 0.71 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58
method | result | size |
default | \(-\frac {2 \left (32 \left (\cos ^{5}\left (f x +e \right )\right )-88 \left (\cos ^{3}\left (f x +e \right )\right )+77 \cos \left (f x +e \right )\right )}{231 f \sin \left (f x +e \right )^{\frac {11}{2}} \sqrt {b \sec \left (f x +e \right )}}\) | \(53\) |
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Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\frac {2 \, {\left (32 \, \cos \left (f x + e\right )^{6} - 88 \, \cos \left (f x + e\right )^{4} + 77 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{231 \, {\left (b f \cos \left (f x + e\right )^{6} - 3 \, b f \cos \left (f x + e\right )^{4} + 3 \, b f \cos \left (f x + e\right )^{2} - b f\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\text {Timed out} \]
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Time = 6.07 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx=\frac {{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,\sqrt {\frac {b}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,992{}\mathrm {i}}{231\,b\,f}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,608{}\mathrm {i}}{231\,b\,f}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,320{}\mathrm {i}}{231\,b\,f}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,64{}\mathrm {i}}{231\,b\,f}\right )\,1{}\mathrm {i}}{32\,{\sin \left (e+f\,x\right )}^{11/2}} \]
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