Integrand size = 21, antiderivative size = 65 \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 b^5}{13 f (b \sec (e+f x))^{13/2}}+\frac {4 b^3}{9 f (b \sec (e+f x))^{9/2}}-\frac {2 b}{5 f (b \sec (e+f x))^{5/2}} \]
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Time = 0.04 (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} \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 b^5}{13 f (b \sec (e+f x))^{13/2}}+\frac {4 b^3}{9 f (b \sec (e+f x))^{9/2}}-\frac {2 b}{5 f (b \sec (e+f x))^{5/2}} \]
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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^{15/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^5 \text {Subst}\left (\int \left (\frac {1}{x^{15/2}}-\frac {2}{b^2 x^{11/2}}+\frac {1}{b^4 x^{7/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = -\frac {2 b^5}{13 f (b \sec (e+f x))^{13/2}}+\frac {4 b^3}{9 f (b \sec (e+f x))^{9/2}}-\frac {2 b}{5 f (b \sec (e+f x))^{5/2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {b (-551+340 \cos (2 (e+f x))-45 \cos (4 (e+f x)))}{2340 f (b \sec (e+f x))^{5/2}} \]
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Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {2 \left (45 \left (\cos ^{6}\left (f x +e \right )\right )-130 \left (\cos ^{4}\left (f x +e \right )\right )+117 \left (\cos ^{2}\left (f x +e \right )\right )\right )}{585 f b \sqrt {b \sec \left (f x +e \right )}}\) | \(50\) |
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Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (45 \, \cos \left (f x + e\right )^{7} - 130 \, \cos \left (f x + e\right )^{5} + 117 \, \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{585 \, b^{2} f} \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (45 \, b^{4} - \frac {130 \, b^{4}}{\cos \left (f x + e\right )^{2}} + \frac {117 \, b^{4}}{\cos \left (f x + e\right )^{4}}\right )} b}{585 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {13}{2}}} \]
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Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.34 \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (45 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{6} - 130 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{4} + 117 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{2}\right )}}{585 \, b^{8} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^5}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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