Integrand size = 21, antiderivative size = 65 \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 65, 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} \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]
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Rule 276
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {\left (-1+\frac {x^2}{b^2}\right )^2}{x^{13/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^5 \text {Subst}\left (\int \left (\frac {1}{x^{13/2}}-\frac {2}{b^2 x^{9/2}}+\frac {1}{b^4 x^{5/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = -\frac {2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {b (-415+180 \cos (2 (e+f x))-21 \cos (4 (e+f x)))}{924 f (b \sec (e+f x))^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\frac {2 \left (21 \left (\cos ^{5}\left (f x +e \right )\right )-66 \left (\cos ^{3}\left (f x +e \right )\right )+77 \cos \left (f x +e \right )\right )}{231 f \sqrt {b \sec \left (f x +e \right )}}\) | \(45\) |
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 66 \, \cos \left (f x + e\right )^{4} + 77 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{231 \, b f} \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 \, {\left (21 \, b^{4} - \frac {66 \, b^{4}}{\cos \left (f x + e\right )^{2}} + \frac {77 \, b^{4}}{\cos \left (f x + e\right )^{4}}\right )} b}{231 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {11}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.31 \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 66 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3} + 77 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )\right )}}{231 \, b^{6} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^5}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
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