Integrand size = 21, antiderivative size = 85 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
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Time = 0.03 (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} \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
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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^{15/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^7 \text {Subst}\left (\int \left (-\frac {1}{x^{15/2}}+\frac {3}{b^2 x^{11/2}}-\frac {3}{b^4 x^{7/2}}+\frac {1}{b^6 x^{3/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac {2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac {6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {(-8939 \cos (e+f x)+887 \cos (3 (e+f x))-155 \cos (5 (e+f x))+15 \cos (7 (e+f x))) \sqrt {b \sec (e+f x)}}{6240 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(444\) vs. \(2(71)=142\).
Time = 1.01 (sec) , antiderivative size = 445, normalized size of antiderivative = 5.24
method | result | size |
default | \(\frac {\left (60 \left (\cos ^{7}\left (f x +e \right )\right )-260 \left (\cos ^{5}\left (f x +e \right )\right )+468 \left (\cos ^{3}\left (f x +e \right )\right )-195 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )+195 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-195 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+195 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-780 \cos \left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{390 f}\) | \(445\) |
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Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.66 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, \cos \left (f x + e\right )^{7} - 65 \, \cos \left (f x + e\right )^{5} + 117 \, \cos \left (f x + e\right )^{3} - 195 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{195 \, f} \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.74 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, b^{6} - \frac {65 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {117 \, b^{6}}{\cos \left (f x + e\right )^{4}} - \frac {195 \, b^{6}}{\cos \left (f x + e\right )^{6}}\right )} b}{195 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {13}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\frac {2 \, {\left (15 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{6} - 65 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{4} + 117 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{2} - 195 \, \sqrt {b \cos \left (f x + e\right )} b^{6}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{195 \, b^{6} f} \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^7(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^7\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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