Integrand size = 21, antiderivative size = 43 \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac {2 b}{7 f (b \sec (e+f x))^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 43, 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} \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac {2 b}{7 f (b \sec (e+f x))^{7/2}} \]
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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^{13/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^3 \text {Subst}\left (\int \left (-\frac {1}{x^{13/2}}+\frac {1}{b^2 x^{9/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac {2 b}{7 f (b \sec (e+f x))^{7/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {\cos ^4(e+f x) (-15+7 \cos (2 (e+f x))) \sqrt {b \sec (e+f x)}}{77 b^3 f} \]
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Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\frac {2 \left (\cos ^{5}\left (f x +e \right )\right )}{11}-\frac {2 \left (\cos ^{3}\left (f x +e \right )\right )}{7}}{f \,b^{2} \sqrt {b \sec \left (f x +e \right )}}\) | \(40\) |
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 \, {\left (7 \, \cos \left (f x + e\right )^{6} - 11 \, \cos \left (f x + e\right )^{4}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{77 \, b^{3} f} \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 \, {\left (7 \, b^{2} - \frac {11 \, b^{2}}{\cos \left (f x + e\right )^{2}}\right )} b}{77 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {11}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.49 \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {2 \, {\left (7 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{5} - 11 \, \sqrt {b \cos \left (f x + e\right )} b^{5} \cos \left (f x + e\right )^{3}\right )}}{77 \, b^{8} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^3}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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