Integrand size = 21, antiderivative size = 87 \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac {6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 87, 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} \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac {6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac {2 b}{3 f (b \sec (e+f x))^{3/2}} \]
[In]
[Out]
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^{17/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^7 \text {Subst}\left (\int \left (-\frac {1}{x^{17/2}}+\frac {3}{b^2 x^{13/2}}-\frac {3}{b^4 x^{9/2}}+\frac {1}{b^6 x^{5/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac {6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac {6 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 = 0.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {b (-7410+4035 \cos (2 (e+f x))-798 \cos (4 (e+f x))+77 \cos (6 (e+f x)))}{18480 f (b \sec (e+f x))^{3/2}} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\frac {2 \left (\cos ^{7}\left (f x +e \right )\right )}{15}-\frac {6 \left (\cos ^{5}\left (f x +e \right )\right )}{11}+\frac {6 \left (\cos ^{3}\left (f x +e \right )\right )}{7}-\frac {2 \cos \left (f x +e \right )}{3}}{f \sqrt {b \sec \left (f x +e \right )}}\) | \(55\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.70 \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 \, {\left (77 \, \cos \left (f x + e\right )^{8} - 315 \, \cos \left (f x + e\right )^{6} + 495 \, \cos \left (f x + e\right )^{4} - 385 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{1155 \, b f} \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 \, {\left (77 \, b^{6} - \frac {315 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {495 \, b^{6}}{\cos \left (f x + e\right )^{4}} - \frac {385 \, b^{6}}{\cos \left (f x + e\right )^{6}}\right )} b}{1155 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {15}{2}}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.24 \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\frac {2 \, {\left (77 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{7} - 315 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{5} + 495 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{3} - 385 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )\right )}}{1155 \, b^{8} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^7(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^7}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
[In]
[Out]