Integrand size = 21, antiderivative size = 41 \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {2 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
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Time = 0.03 (sec) , antiderivative size = 41, 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, 14} \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {2 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
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Rule 14
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {b^3 \text {Subst}\left (\int \frac {-1+\frac {x^2}{b^2}}{x^{7/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^3 \text {Subst}\left (\int \left (-\frac {1}{x^{7/2}}+\frac {1}{b^2 x^{3/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {2 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {(-17 \cos (e+f x)+\cos (3 (e+f x))) \sqrt {b \sec (e+f x)}}{10 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(35)=70\).
Time = 0.26 (sec) , antiderivative size = 425, normalized size of antiderivative = 10.37
method | result | size |
default | \(-\frac {\left (5 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-5 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-4 \left (\cos ^{3}\left (f x +e \right )\right )+5 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-5 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+20 \cos \left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{10 f}\) | \(425\) |
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Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {2 \, {\left (\cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{5 \, f} \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {2 \, {\left (b^{2} - \frac {5 \, b^{2}}{\cos \left (f x + e\right )^{2}}\right )} b}{5 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {5}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\frac {2 \, {\left (\sqrt {b \cos \left (f x + e\right )} b^{2} \cos \left (f x + e\right )^{2} - 5 \, \sqrt {b \cos \left (f x + e\right )} b^{2}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{5 \, b^{2} f} \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^3(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^3\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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