Integrand size = 21, antiderivative size = 63 \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {2 b^5}{9 f (b \sec (e+f x))^{9/2}}+\frac {4 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 63, 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} \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {2 b^5}{9 f (b \sec (e+f x))^{9/2}}+\frac {4 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \]
[In]
[Out]
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^{11/2}} \, dx,x,b \sec (e+f x)\right )}{f} \\ & = \frac {b^5 \text {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {2}{b^2 x^{7/2}}+\frac {1}{b^4 x^{3/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f} \\ & = -\frac {2 b^5}{9 f (b \sec (e+f x))^{9/2}}+\frac {4 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac {2 b}{f \sqrt {b \sec (e+f x)}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {(554 \cos (e+f x)-47 \cos (3 (e+f x))+5 \cos (5 (e+f x))) \sqrt {b \sec (e+f x)}}{360 f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(53)=106\).
Time = 0.21 (sec) , antiderivative size = 435, normalized size of antiderivative = 6.90
method | result | size |
default | \(-\frac {\left (20 \left (\cos ^{5}\left (f x +e \right )\right )+45 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-45 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )-72 \left (\cos ^{3}\left (f x +e \right )\right )+45 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-45 \ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+180 \cos \left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{90 f}\) | \(435\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {2 \, {\left (5 \, \cos \left (f x + e\right )^{5} - 18 \, \cos \left (f x + e\right )^{3} + 45 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{45 \, f} \]
[In]
[Out]
Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79 \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {2 \, {\left (5 \, b^{4} - \frac {18 \, b^{4}}{\cos \left (f x + e\right )^{2}} + \frac {45 \, b^{4}}{\cos \left (f x + e\right )^{4}}\right )} b}{45 \, f \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {9}{2}}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=-\frac {2 \, {\left (5 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{4} - 18 \, \sqrt {b \cos \left (f x + e\right )} b^{4} \cos \left (f x + e\right )^{2} + 45 \, \sqrt {b \cos \left (f x + e\right )} b^{4}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{45 \, b^{4} f} \]
[In]
[Out]
Timed out. \[ \int \sqrt {b \sec (e+f x)} \sin ^5(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^5\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
[In]
[Out]