Integrand size = 21, antiderivative size = 85 \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {2 b^7}{9 f (b \sec (e+f x))^{9/2}}-\frac {6 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {6 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \]
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Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2702, 276} \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {2 b^7}{9 f (b \sec (e+f x))^{9/2}}-\frac {6 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {6 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \]
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Rule 276
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {b^7 \text {Subst}\left (\int \frac {\left (-1+\frac {x^2}{b^2}\right )^3}{x^{11/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^7 \text {Subst}\left (\int \left (-\frac {1}{x^{11/2}}+\frac {3}{b^2 x^{7/2}}-\frac {3}{b^4 x^{3/2}}+\frac {\sqrt {x}}{b^6}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {2 b^7}{9 f (b \sec (e+f x))^{9/2}}-\frac {6 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {6 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.61 \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {b (2366+1803 \cos (2 (e+f x))-78 \cos (4 (e+f x))+5 \cos (6 (e+f x))) (b \sec (e+f x))^{3/2}}{720 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(71)=142\).
Time = 0.87 (sec) , antiderivative size = 446, normalized size of antiderivative = 5.25
\[\frac {\sqrt {b \sec \left (f x +e \right )}\, b^{2} \left (20 \left (\cos ^{5}\left (f x +e \right )\right )+135 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-135 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-108 \left (\cos ^{3}\left (f x +e \right )\right )+135 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-135 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+540 \cos \left (f x +e \right )+60 \sec \left (f x +e \right )\right )}{90 f}\]
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Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.82 \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (5 \, b^{2} \cos \left (f x + e\right )^{6} - 27 \, b^{2} \cos \left (f x + e\right )^{4} + 135 \, b^{2} \cos \left (f x + e\right )^{2} + 15 \, b^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{45 \, f \cos \left (f x + e\right )} \]
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Timed out. \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.78 \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}} + \frac {5 \, b^{6} - \frac {27 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {135 \, b^{6}}{\cos \left (f x + e\right )^{4}}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {9}{2}}}\right )} b}{45 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (5 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{4} - 27 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{2} + 135 \, \sqrt {b \cos \left (f x + e\right )} b^{4} + \frac {15 \, b^{5}}{\sqrt {b \cos \left (f x + e\right )} \cos \left (f x + e\right )}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{45 \, b^{2} f} \]
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Timed out. \[ \int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^7\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2} \,d x \]
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